Introduction to Maxima for Economicsleydold/maxima/MaximaSkript.pdf · 2011. 9. 26. · 2.3 Maxima...

Introduction to Maxima for Economics

Josef Leydold Martin Petry

September 26, 2011

Institute for Statistics and Mathematics, WU Wien

The following verions of the Maxima CAS has been used when writing this manual:

Maxima 5.25.1wxMaxima 11.04.0

© 2010–2011 Josef Leydold

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike3.0 Austria License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/at/deed.en or send a letter to Creative Commons, 171 SecondStreet, Suite 300, San Francisco, California, 94105, USA.

http://creativecommons.org/licenses/by-nc-sa/3.0/at/deed.en

http://creativecommons.org/licenses/by-nc-sa/3.0/at/deed.en

Contents

1 Introduction 11.1 Maxima Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 How to Read this Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Basics 32.1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 wxMaxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Maxima as a Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Basic Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Variables and User Defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 Use Output for Further Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 System Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.11 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.12 Numeric Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.13 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.14 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.15 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.16 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.17 Loading Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.18 Writing Your Own Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.19 LISP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Symbolic Computation 213.1 Symbolic Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Expand, Factor, and Simplify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Verbs and Nouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Conditional Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Assumptions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 Local Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Plotting 314.1 Plots in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Plots in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Plot Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 The Draw Package 455.1 Draw Scenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Graphic Objects in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 Graphic Objects in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.4 Graphic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Linear Algebra 656.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

iii

Contents

6.3 Operations on Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.4 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.5 Lists (Revised) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.6 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.7 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7 Calculus 757.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.3 Derivatives and Total Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.4 Dependencies and Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.5 Jacobian Matrix and Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.6 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.7 Minima and Maxima of Univariate Functions . . . . . . . . . . . . . . . . . . . . . . . 827.8 Minima and Maxima of Multivariate Functions . . . . . . . . . . . . . . . . . . . . . . 837.9 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8 Ordinary Differential Equations (ODE) 898.1 Analytic Solutions of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . 898.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.3 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9 Economic Examples 979.1 Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.2 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.3 Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.4 Cournot Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019.5 Cobb-Douglas Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019.6 Stationary Points and Cobb-Douglas Function . . . . . . . . . . . . . . . . . . . . . . 1029.7 Constraint Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Answers to the Exercises 107

Index 113

iv

1 Introduction

Maxima is a computer algebra system (CAS) like Maple® and Mathematica®. Thus it is a powerfultool for the manipulation of symbolic and numerical expressions, including differentiation, inte-gration, vectors, matrices, . . . and so on. It also provides commands for plotting functions, curvesand data in two and three dimensions. In opposition to other CAS Maxima has a quite long historyand is open source software. From the Maxima manual:

Maxima is a descendant of Macsyma, the legendary computer algebra system devel-oped in the late 1960s at the MIT. It is the only system based on that effort still publiclyavailable and with an active user community. Macsyma was revolutionary in its day,and many other systems were inspired by it. In 1998, Professor W. F. Schelter of theUniversity of Texas obtained permission form the Department of Energy to release theMacsyma source code under the GNU Public License, and in 2000 he initiated the Max-ima project at SourceForge to maintain and develop Macsyma, now called Maxima.

Maxima is available on many operating systems including MS Windows®, Mac OSX and Linux.The software can be downloaded from

http://maxima.sourceforge.net/.

There one can find links to more information as well as to related software.

Maxima also allows to define your own commands. Thus it is easy to Maxima’s capabilities andadjust it to own’s need. Thus you find contributed “packages” that have been developed and madepublicly available by users of this CAS.

The obvious advantage of open source software (OSS) is that it can be downloaded and used forfree. Nevertheless, there are also other gains for users: It allows to see how other people solveda similar problem (and you are free to copy and modify their solutions). In case of unexpectedresults one can investigate the sources and thus there is some chance to detect a possible error1.

1.1 Maxima Interfaces

Maxima at its heart has a command line interface and by itself it is not capable of displaying for-matted mathematics beyond the plain text level. For most users that has been grown up withgraphical user interfaces (GUI) this is unfamiliar and may seem quite arcane. Fortunately, nowa-days more fancy GUIs are available.

We will use the most popular one called wxMaxima. It is available at

http://wxmaxima.sourceforge.net/.

Its interface consists of a notebook with menus and a toolbar which allows an inexperienced userto pick an appropriate command. These commands as well as their outputs are printed into thenotebook. On the other hand these commands can also directly be entered and edited and thusprovide a powerful workspace for the experienced user. The notebook can then be saved at the

1The core functionality of Maxima is written in LISP.

1

http://maxima.sourceforge.net/

http://wxmaxima.sourceforge.net/

1 Introduction

end of a session and reloaded for later use. Thus it allows the user to construct a document thatconsists of text, calculations, and plots.

In this tutorial we simulate the output as it is given by wxMaxima. Moreover, we will explain somefeatures of this interface. However, we mainly describe the commandline interface of Maxima.Exploring the toolbar is left to the reader.

Alternatives GUIs are Xmaxima which is shipped with Maxima. It is less fancy but more stablethan wxMaxima that is currently being actively developed. A nice interface is also provided byTexmacs, a WYSIWYG word processing system that allows to include Maxima code and prints itsoutput. Last but not least, an really excellent Emacs mode has been written for Maxima.

1.2 How to Read this Tutorial

The sections in each chapter first shows a table of (new) Maxima commands.

Command Description

cmd a command

In addition we may need to change some special variables that control the behavior of Maxima’soutput or evaluations. These are listed in the following way.

Variable Default Description

var true a special variable

Examples of Maxima code are printed in a yellow box. The output mimics the output of the wx-Maxima program.

(%i1) input;

(%o1) output

Maxima code is running text is printed using courier.

Keyboard buttons are shown as KEY . Notice that SHIFT + ENTER means that one has first press andhold the SHIFT button and then press the ENTER button.

ImportantCaveats and possible errors are marked by an exclamation mark inside a triangle.BEach Section closes with a small set of exercises.

The tutorial organized in the following way. Chapter 2 presents the basic commands and conceptthat are necessary to run the Maxima system. Here we mainly focus on numerical calculationsbut we also point to possible errors. In Chapter 3 we discuss the principles of symbolic computa-tions. The remaining chapters then deal with special topics, like Plotting (Chap. 4), Linear Algebra(Chap. 6), and Calculus (Chap. 7).

The topics in this manual are strongly influenced by typical applications in economics. Thus wepresent some relevant examples in Chapter 9.

ImportantThis manual gives insight into a small subset of the Maxima system. In addition, we tried to makeBthe presentation as simple as possible and thus we did not explain all the details. Thus the readeris encouraged to read the online documentation (see Sect. 2.9) for a comprehensive description ofthe presented commands.

2

2 Basics

Maxima is a computer algebra system that accepts some input, evaluates the given commandsand returns the results. As with all CAS we have to be familiar with the syntactical rules of thissystem.

2.1 Getting Started

Symbol Description

%i1 Input (input prompt)%o1 Output of evaluation (output prompt); End-of-input marker$ End-of-input marker. Output will be suppressed

Start Maxima or wxMaxima. Then you can enter your commands that must be terminated by asemicolon ; .Observe that your input will be automatically prefixed by %i1. The given string is Bthen evaluated by Maxima according to rules that are basically based on pattern matching. Theresult is then printed and the output is prefixed by %o1. It can be used to refer to the result infurther computations, see Sect. 2.8 below.

A command may also be terminated by the special symbol $ instead of a semicolon. Then theoutput of your result is suppressed, i.e., Maxima evaluates your input expression but does notshow its results. This can be quite convenient when an intermediate result is very long and fillsseveral pages. Here is a simple example.

(%i1) 2+3;

(%o1) 5

(%i2) 4*5$

As you can observe your input and output with be enumerated. It is also possible to put more thanone command on one line. On the other hand an command can also be spread over two or morelines (thus the command has to be terminated by means of ; or $).

(%i3) 2+3; 4*5;

(%o3) 5

(%o4) 20

(%i5) 2+3*(4+5)+6*7-8/4;

(%o5) 69

3

2 Basics

2.2 wxMaxima

wxMaxima provides a more convenient interface. It shows a notebook where one can insert anexpression. This notebook is organized using co called cells. Notice that the bar on the left handside indicates which input and output cells belong together. Whenever, a cell is currently editedor evaluated the color of this bar changes. (Some evaluations can be very time consuming.)

ImportantOne has to press SHIFT + ENTER to start the evaluation of your input. A missing ; character is thenBautomatically inserted. Notice that pressing ENTER alone (without pressing SHIFT ) only inserts aline break1 even when a ; or $ is present.

wxMaxima also provides a toolbar where one can select commands from menus. This is veryconvenient for the beginner as it disburdens one from memorizing the command syntax. (But itis cumbersome for the more experienced user.)

Using the notebook one can insert his or her commands almost everywhere on the page. Even fur-ther command can be edited and reevaluated. (Of course the number of the cell is then updated.)Thus the cells are not necessary in chronological order.

In addition one can insert sections headers and text to comment one’s calculations. For this pur-pose use the corresponding entries in menu Cell.

When a Maxima session is finished one can save a notebook (using File->Save) and reload it again(using File->Open). Here is a typical wxMaxima notebook.

1However, this default behavior of wxMaxima can be changed: tick the box Enter evaluates cells in the wxMaxima con-figuration window.

4

2.3 Maxima as a Calculator

2.3 Maxima as a Calculator

Almost any mathematical expression is available in Maxima. The operators are the same as onyour calculator, see Sections 2.4 and 2.6.

2.4 Basic Arithmetic Operators

Operator Description

+ Addition- Subtraction/ Division* Multiplication^ Raise to power

Some examples are:

(%i1) 5+3;

(%o1) 8

(%i2) 22-7;

(%o2) 15

(%i3) 121/11;

(%o3) 11

(%i4) 25*4;

(%o4) 100

(%i5) 2^8;

(%o5) 256

(%i6) (2+3)^4-(5+6+7)/(2+4);

(%o6) 622

Maxima can compute very large numbers as the following example demonstrates.

(%i7) 2^100;

(%o7) 1267650600228229401496703205376

2.5 Constants

Constant Description

%e Euler’s number e = 2.71828 . . .%pi π = 3.14159 . . .%i Imaginary unit i =

√−1

inf Positive infinity ∞minf Minus infinity−∞

Maxima knows important numerical constants like e and π as well as±∞.

5

2 Basics

2.6 Functions

Function Description

abs(x) Absolute value of xsqrt(x) Square root of xlog(x) Natural logarithm (i.e, to base e) of xexp(x) Exponential function of xsin(x) Sine of xcos(x) Cosine of xtan(x) Tangent of x

As any calculator Maxima provides commands for computing functions.

(%i1) cos(%pi);

(%o1) −1

(%i2) exp(2);

(%o2) %e2

(%i3) abs(-5);

(%o3) 5

ImportantNotice that the arguments to sin and cosmust be angles given in radians (i.e., where a right angleBis π

2 ). If you need to do computations using degrees (i.e., where a right angle is 90◦), then you have

to convert degree d into radian x using x = dπ

180.

The exponential function for an arbitrary base a can computed by means of the the ^ operator.Similarly the n-th root can be computed using ^(1/n). Notice that %e^x is an alternative forexp(x).

(%i4) 10^4;

(%o4) 1000

(%i5) 1000^(1/3);

(%o5) 10

(%i6) 8^(2/3);

(%o6) 4

Maxima only provides the natural logarithm function (with base e). The common logarithm (base10) can be computed using

log10(x) =log(x)log(10)

.

ImportantDo not use ln(x) to compute the natural logarithm. Maxima does not know this function andBthus it returns it unevaluated.

(%i7) log(%e);

(%o7) 1

(%i8) ln(%e);

(%o8) ln(%e)

6

2.7 Variables and User Defined Functions

2.7 Variables and User Defined Functions

operator Description

: Assignment operator:= Function definition operatorkill(x) Removes all bindings (value, function) from xkill(all) Removes all bindings from all items


values [] List of user-defined variablesfunctions [] List of user-defined functions

Expressions like numbers can be stored in variables, i.e., a sequence of alphanumeric characters(that start with a letter). The alphanumeric characters are A through Z, a through z, 0 through 9,%, and _. Examples of valid variable names are: a, beta, or y_1. Thus one can store the resultof any evaluation by means of the assignment operator :. This operator evaluates its right-handside and associates that value with the left-hand side. When the variable is evaluated in furthercomputations, then it is replaced by its value. Here is a simple example.

(%i1) x;

(%o1) x

(%i2) x: 2+3;

(%o2) 5

(%i3) x;

(%o3) 5

(%i4) y: 2*x^2;

(%o4) 50

ImportantMaxima is case-sensitive, that is, the identifiers foo, FOO, and Foo are distinct. BIt also possible to extend Maxima with one’s own functions by means of the function definitionoperator :=. Function names are similar to variable names but are followed by parenthesis (...)that contain a comma separated list of its arguments (which themselves are variables). The right-hand side of the function assignment operator := (i.e., the function body) is never evaluated.

In the following example we define functions sqr for computing the square of a given expressionand sindeg for the sine function where the angle is given in degrees. Notice the function bodycontains expression x which is note evaluated. Otherwise it would be replaced by value 5 whichhas been assigned in the previous example.

(%i5) sqr(x):= x^2;

(%o5) sqr(x) := x2

(%i6) sqr(4);

(%o6) 16

(%i7) sindeg(x):= sin(x*%pi/180);

(%o7) sindeg(x) := sin( xπ

180

)(%i8) sindeg(45);

(%o8)1√2

7

2 Basics

Of course it is possible to define functions in two or more variables.

(%i9) norm(x,y):= sqrt(x^2+y^2);

(%o9) norm(x, y) :=√

x2 + y2

(%i10) norm(3,4);

(%o10) 5

The system variables values and functions contain a list of user defined variables and func-tions, resp.

(%i11) values;

(%o11) [x, y]

(%i12) functions;

(%o12) [sindeg(x), sqr(x), norm(x, y)]

Notice that both variables and functions remain persistent until the Maxima session is closed.Sometimes it is convenient to remove these again. This can be accomplished by function kill.

(%i13) kill(x);

(%o13) done

(%i14) values;

(%o14) [y]

(%i15) functions;

(%o15) [sindeg(x), sqr(x), norm(x, y)]

(%i16) kill(all);

(%o16) done

(%i17) values;

(%o17) [ ]

(%i18) functions;

(%o18) [ ]

ImportantYou must not use symbol = to define variables and functions. This symbol is used for definingBequations, see Sect. 2.15.

(%i19) x=3;

(%o19) x = 3

(%i20) x;

(%o20) x

8

2.8 Use Output for Further Computations

2.8 Use Output for Further Computations

Command Description

% Result of last evaluation%oi Result of i-th evaluation (content of output cell i)

The operator % refers to the output expression most recently computed my Maxima, whether ornot it was displayed. In addition the result of the i-th computation is available by %oi.

(%i1) 2+3;

(%o1) 5

(%i2) % * 20;

(%o2) 100

(%i3) % - 10;

(%o3) 90

(%i4) %o1 - 10;

(%o4) −5

ImportantNotice that % refers to the result of the last evaluation. As you can enter a command almost ev- Berywhere in a wxMaxima notebook this is not necessarily the content of the output cell just aboveyour current input cell.

2.9 Help

Command Description

? cmd Show help page for command cmd (short for describe)?? cmd Search for inexact matches of cmd in help pagesapropos("cmd") List of all Maxima names that contain substring cmd

Help pages for all commands and operators are available using the special symbols ? cmd and?? cmd. The double question mark ?? can be used to search for string cmd in the manual.

(%i1) ? sqrt;-- Function: sqrt (<x>)

The square root of <x>. It is represented internally by`<x>^(1/2)'. See also `rootscontract'.`radexpand' if `true' will cause nth roots of factors of a productwhich are powers of n to be pulled outside of the radical, e.g.`sqrt(16*x^2)' will become `4*x' only if `radexpand' is `true'.

There are also some inexact matches for `sqrt'.Try `?? sqrt' to see them.

ImportantIt is important to insert a space between ? and cmd. Otherwise the input is interpreted as an Binternal LISP variable which is simply returned. It also might result in a weird error message fromthe LISP interpreter.

(%i2) ?sqrt;

(%o2) sqrt

9

2 Basics

If you only remember a substring of a command name, then apropos is quite useful. It returns alist of all those Maxima names that contain this string.

(%i3) apropos("sqrt");

(%o3) [isqrt, sqrt, sqrtdisp f lag]

For an alternative method to access the manual within wxMaxima press the F1 button or use theHelp button in the menu bar. It gives access to the whole library including an index and a searchfunction.

2.10 System Variables

Command Description

ev(x,envir) Evaluate expression x with in given environment envirreset() Resets many (global) system variables

Maxima uses a set of system variables to control the behavior of the system. For example, inSect. 2.11 below variable fpprintprec is used to control the printing of floating point numbers,and numer controls whether mathematical functions are evaluated in floating point or not.

One may use assign operator : to change value of system variables globally. Command reset()allows to reset many global system variables and some other variables, to their default values.

An alternative approach to changing and resetting system variables is the use of command evwhich allows to evaluate an expression with locally changed system variables. An example withvariable numer is given in Sect. 2.12 below.

2.11 Numbers

Command Description

123 integer 123123. integer 123123.0 floating point number 123123e0 floating point number 123123b0 big float number 123


fpprec 16 Number of significant digits for arithmetic of bigfloat numbersfpprintprec 0 Number of digits when printing a float (default: all significant digits)

Maxima distinguishes between four different types of numbers:

• integers: . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

• rational numbers: ratio of two integers

• floating point numbers: consist of a mantissa of (approximately) 16 decimal digits and an ex-ponent to base 10, e.g., 1.234567890123456 · 105. In common speech these are called decimalnumbers and usually written without the exponent; in our example 123456.7890123456.

• arbitrary precision floating point numbers (called bigfloat numbers: floating point numberswhere the size of the mantissa can be set to some fixed but arbitrary number (using the sys-tem variable fpprec.

10

2.11 Numbers

In addition, Maxima can handle some special constants, see Sect. 2.5.

Floating point numbers can be entered either as a decimal number with at least one digit after thedecimal point, e.g., 123.0, or using the scientific notation, e.g., 123e0.

Integers and rational numbers are stored without loss of precision while this is not possible forfloating point numbers. They can be seen as an approximation to real numbers. Notice that thedecimal expression of real numbers (as elements from R) may have an infinite number of digitsas in 1

7 = 0.1428571428571428 . . . or√

2 = 1.414213562373095 . . . . When stored as floating pointnumbers only a limited number of digits can be stored and one looses precision. Additions andsubtractions of floating point numbers then may results in further loss of precision due to cancel-lation errors. The following example demonstrates the difference between rational numbers andfloating point numbers.

(%i1) 2/10 * 11 - 2 - 4/20;

(%o1) 0

(%i2) 2.0/10.0 * 11.0 - 2.0 - 4.0/20.0;

(%o2) 1.665334536937735 10−16

ImportantIn opposition to decimal representation modern computer use a binary representation of num- Bbers by means of the binary digits 0 and 1 called bits. As a consequence numbers like 0.2 cannotbe stored exactly due an infinite binary representation.

Maxima tries to do all its evaluation as exact as possible. Consequently, Maxima reduces rationalnumbers or simplifies numerical expression where possible but does not convert to floating pointnumbers unless forced to do so (see Sect. 2.12 below). In particular Maxima also returns specialnumbers as results of computations.

(%i3) 19 / 3;

(%o3)193

(%i4) 2^1000;

(%o4) 107150860718626732094842504906[242 digits]429831652624386837205668069376

(%i5) sqrt(2);

(%o5)√

2

(%i6) 14 / 6;

(%o6)73

(%i7) sqrt(12);

(%o7) 3√

2

(%i8) sqrt(8);

(%o8) 23/2

(%i9) exp(3);

(%o9) %e3

(%i10) atan(1);

(%o10)π

4(%i11) tan(%pi/4);

(%o11) 1

11

2 Basics

When we use floating point numbers instead, we get a less precise result, i.e., stored as floatingpoint numbers. As the first line of the following example shows, it is often sufficient to insert justone floating point number in order to obtain a floating point answer. As the last line shows, theanswer can sometimes a bit weird.

(%i12) 19.0 / 3;

(%o12) 6.333333333333333

(%i13) 2.0^1000;

(%o13) 1.07150860718627 10301

(%i14) sqrt(2.0);

(%o14) 1.414213562373095

(%i15) sqrt(12.0);

(%o15) 3.464101615137754

(%i16) exp(3.0);

(%o16) 20.08553692318767

(%i17) atan(1.0);

(%o17) .7853981633974483

(%i18) tan(%pi/4.0);

(%o18) tan(0.25 ∗%pi)

We again stress that the difference between integers and floating point numbers is important. Inthe inputs in the following two lines of code are different and thus are handled differently!B(%i19) 123;

(%o19) 123

(%i20) 123.0;

(%o20) 123.0

Sometimes it can be annoying when 16 digits of floating point numbers are printed. This can becontrolled by setting system variable fpprintprec.

(%i21) fpprintprec: 4$

(%i22) sqrt(2.0);

(%o22) 1.141

(%i23) fpprintprec: 0$

(%i24) sqrt(2.0);

(%o24) 1.414213562373095

2.12 Numeric Results

Command Description

float(x) Convert x to floating point numberbfloat(x) Convert x to big floatset_display(ascii) Set display mode to plain ASCIIset_display(xml) Set display mode to XML (default)

12

2.12 Numeric Results


numer false if true, then mathematical functions are evaluated in floating point

When we are interested in numeric results we can set the system variable numer to true. Thenall evaluations are performed in floating point numbers. This can also be done locally using thecommand ev.

(%i1) ev(19/3, numer:true);

(%o1) 6.333333333333333

In many situations we can simplify the notation. numer is interpreted as numer:true. Thus evmay be omitted, too. The following works as well.

(%i2) 19/3, numer;

(%o2) 6.333333333333333

(%i3) sin(4), numer;

(%o3) −0.75680249530793

(%i4) %pi, numer;

(%o4) 3.141592653589793

An alternative approach is to use the float command.

(%i5) float(19/3);

(%o5) 6.333333333333333

(%i6) float(sin(4));

(%o6) −0.75680249530793

(%i7) float(%pi);

(%o7) 3.141592653589793

wxMaxima tries to print Maxima’s output in a nice manner. Number are printed into one line.Suppose you need all digits of 100! or the first 500 digits of π. Then not all digits are displayedwhich may not be what you want.

(%i8) 100!;

(%o8) 933262154439441526816992388562[98 digits]916864000000000000000000000000

(%i9) fpprec: 500$

(%i10) bfloat(%pi);

(%o10) 3.1415926535897932384626433832[443 digits]8857527248912279381830119491b0

(%i11) reset()$

However, it is possible to switch back to Maxima’s native output format using commandset_display(ascii). Then all digits are printed albeit in a not so nice form. The backslash sign\ at the end of each line indicates that it is continued on the next. Do not forget to reset the displayformat again by means of set_display(xml);

13

2 Basics

(%i12) set_display(ascii)$

(%i13) 100!;

(%o13) 93326215443944152681699238856266700490715968264381621468592963895217599\993229915608941463976156518286253697920827223758251185210916864000000000000000\000000000

(%i14) fpprec: 500$

(%i15) bfloat(%pi);

(%o15) 3.141592653589793238462643383279502884197169399375105820974944592307816\406286208998628034825342117067982148086513282306647093844609550582231725359408\128481117450284102701938521105559644622948954930381964428810975665933446128475\648233786783165271201909145648566923460348610454326648213393607260249141273724\587006606315588174881520920962829254091715364367892590360011330530548820466521\384146951941511609433057270365759591953092186117381932611793105118548074462379\9627495673518857527248912279381830119491b0

(%i16) reset()$

(%i17) set_display(xml)$

2.13 Lists

Command Description

[...] Marks a list or enclose subscripts of a listlength(L) Length of list Lappend(...) Concatenate given listsdelete(x,L) Remove x from list L

Lists are used to combine several items into a single object. They are displayed by enclosing itselements between square brackets. A simple method to create list by means of square brackets[. . .]. A list may even contain other lists as its entries. It may also consists of one or zero elements.

(%i1) L: [1,4,9,16,25,[36,49]];

(%o1) [1, 4, 9, 16, 25, [36, 49]]

(%i2) emptylist: [];

(%o2) [ ]

An element of a list can be accessed by its index enclosed in square brackets. The first element hasindex 1 and the largest index must not exceed the length of the list.

(%i3) L[4];

(%o3) 16

(%i4) length(L);

(%o4) 6

(%i5) L[3]: -99;

(%o5) −99

(%i6) L;

(%o6) [1, 4,−99, 16, 25, [36, 49]]

14

2.14 Strings

There exist a couple of commands for manipulating lists. Concatenating lists (append elements ofone list to another) and deleting elements from a list are the most important functions.

(%i7) append(L,[1,2,3]);

(%o7) [1, 4,−99, 16, 25, [36, 49], 1, 2, 3]

(%i8) delete(1,L);

(%o8) [4,−99, 16, 25, [36, 49]]

Notice that these commands return a new list and do not modify the existing one. For the latterpurpose one has to replace the original list.

(%i9) L: delete(4,L);

(%o9) [1,−99, 16, 25, [36, 49]]

(%i10) L;

(%o10) [1,−99, 16, 25, [36, 49]]

2.14 Strings

Command Description

"a string" Character string

Character strings are enclosed in double quote marks " for input. Strings may contain any char-acters, including embedded tab, newline, and carriage return characters.

(%i1) s1: "This is a String.";

(%o1) This is a String.

2.15 Equations

Command Description

= Equation operatorsolve(eqn,x) Solve equation eqn with respect to xlhs(eqn) Left-hand side of equation eqnrhs(eqn) Right-hand side of equation eqnallroots(polyn) Numerical approximations of roots of polynomial polyn of one variable

Command solve tries to solve algebraic equations. It returns a list of solution equations. (Usefloat when numeric solutions are required.)

(%i1) solve(x^2+3*x-1=0, x);

(%o1) [x = −√

13 + 32

, x =

√13− 3

2]

(%i2) float(%);

(%o2) [x = −3.302775637731995, x = .3027756377319946]

15

2 Basics

Finding roots and zeros of expression are very important special applications of solving equations.Thus if the expression expr that given as an argument to solve is not an equation, then the equa-tion expr=0 is assumed in its place. Thus we also obtain the roots of the above polynomial by thefollowing command.

(%i3) sol: solve(x^2+3*x-1, x);

(%o3) [x = −√

13 + 32

, x =

√13− 3

2]

The solutions can be reverted into “non-equations” or used in further computations by meansof command ev. An alternative way is to use command rhs to extract the expression from theright-hand side of the equation.

(%i4) ev(x, sol[1]);

(%o4) −√

13 + 32

(%i5) rhs(sol[2]);

(%o5)

√13− 3

2

The variable can also be a more complex expression like a function.

(%i6) solve(exp(f(x))=10^y, f(x));

(%o6) [ f (x) = log(10) y]

Systems of two or more expressions as well as their variables must be encapsulated in lists. Eachsolution is then also returned as list. (Thus we get a list of lists.) Notice that we can assign equa-tions to variables.

(%i7) eq1: 3*x^2-y^2=2;

(%o7) 3 x2 − y2 = 2

(%i8) eq2: x^2+y^2=2;

(%o8) x2 + y2 = 2

(%i9) solve([eq1,eq2], [x,y]);

(%o9) [[x = −1, y = −1], [x = −1, y = 1], [x = 1, y = −1], [x = 1, y = 1]]

solve returns its solution as equations. These solutions can be used in further computations bymeans of ev.

(%i10) s: solve(x^3+2*x^2-5*x-6=0, x);

(%o10) [x = −3, x = −1, x = 2]

(%i11) z: ev(%pi^x, s[2]);

(%o11)1π

Importantsolve does not always find (all) solutions of a system of equations. Then an empty list is returned.BIn the following example the “obvious” solution [x=1,y=1] is not found.

(%i12) solve([x^(-1/2)*y^(1/2)=1, x^(1/2)*y^(-1/2)=1], [x,y]);

(%o12) [ ]

(%i13) ev([x^(-1/2)*y^(1/2)-1, x^(1/2)*y^(-1/2)-1], [x=1,y=1]);

(%o13) [0, 0]

16

2.16 Complex Numbers

In general there are no closed form solutions for the roots of polynomials of degree 5 or larger.Then command allroots may be used to get at least a numerical approximation of the roots.

(%i14) solve(x^5-x^4+2*x^3+x^2-x+5, x);

(%o14) [0 = x5 − x4 + 2 ∗ x3 + x2 − x + 5]

(%i15) allroots(x^5-x^4+2*x^3+x^2-x+5);

(%o15) [x = 1.07479 %i + .970613, x = .970613− 1.07479 %i, x = −1.1828,x = 1.41457 %i + .120788, x = .120788− 1.41457 %i]

2.16 Complex Numbers

Command Description

rectform(x) Express complex number x in Cartesian form a + b ipolarform(x) Express complex number x in polar form r eiθ

realpart(x) Real part of a complex number ximagpart(x) Imaginary part of a complex number xtrigsimp(x) Simplifies expressions that contain trigonometric functions

Maxima performs many of its computation over the field of complex numbers. Thus the solutionsmay contain imaginary numbers.

(%i1) solve(x^2-4*x+13=0,x);

(%o1) [x = 2− 3 %i, x = 3 %i + 2]

While this is quite convenient for many applications it may be annoying when we are interestedin purely real solutions. The situation may become even worse as the following example demon-strates. Here the solution is computed by means of Cardano’s formula. It looks like complex num-bers.

(%i2) s: solve(x^3-x+1/3=0,x);

(%o2) [x =

√3 %i2 − 1

2

3(

%i

2 332− 1

6

) 13+

(%i

2 332− 1

6

) 13(−√

3 %i2− 1

2

),

x =

(%i

2 332− 1

6

) 13(√

3 %i2− 1

2

)+−√

3 %i2 − 1

2

3(

%i

2 332− 1

6

) 13

,

x =

(%i

2 332− 1

6

) 13+

1

3(

%i

2 332− 1

6

) 13]

However, all three solutions are real numbers. We can try to transform these complex numbersinto Cartesian form, i.e., into the representation a + b i where a ∈ R and b ∈ R are called realand imaginary part, respectively. One then can try to reduce the imaginary part to 0 by means ofcommand trigsimp.

17

2 Basics

(%i3) trigsimp(rectform(s));

(%o3) [x =

√3 sin

( 5 π18)− cos

( 5 π18)

√3

, x = −√

3 sin( 5 π

18)+ cos

( 5 π18)

√3

, x =2 cos

( 5 π18)

√3

]

(%i4) float(%);

(%o4) [x = .3949308436346985, x = −1.137158042603258, x = .7422271989685593]

If we do not use trigsimp or if it is not possible to reduce the imaginary part to 0, then a tinyimaginary part might remain when we convert the solutions to floating point numbers due toround-off errors. It can be transformed to a purely real solution by using their real parts. However,we must be confidence that the solution is not a complex number.

(%i5) float(rectform(s));

(%o5) [x = .3949308436346984− 1.110223024625157 10−16 %i,x = −4.163336342344337 10−17 %i− 1.137158042603257, x = .7422271989685593]

(%i6) realpart(%);

(%o6) [x = .3949308436346984, x = −1.137158042603257, x = .7422271989685593]

2.17 Loading Packages

Command Description

load(pkg) Evaluate content of file pkg (“Load package”)

Like many computing environments Maxima provides a basic set of commands (core) and offers avariety of add-on packages for special tasks. This keeps starting time and memory consumptionsat a minimum and allows large functionality. Moreover, it is thus possible to extend the capabilitiesof the system. Such packages are either automatically loaded on demand (by means of a list ofcommands for which packages have to be loaded) or the packages have to be explicitly loaded.For the latter we have to load the corresponding files.

(%i1) load(vect);

(%o1) /usr/share/maxima/5.22.1/share/vector/vect.mac

Notice that the loaded functions are added to the list of user-defined functions and can be dis-played and removed using functions and kill(all), respectively.

2.18 Writing Your Own Package

Command Description

/*...*/ Comment: all characters between these delimiters are ignored

When working with Maxima one may use a set of functions or a complex variable definitions quitefrequently. The code for these functions can be collected in a file and loaded into one’s Maximasession by means of command load as described in Sect. 2.17.

ImportantThe file must have extension .mac.BFor example, start your favorite editor, enter the line

18

2.19 LISP

/* Euclidean norm of vector [x,y] */norm(x,y):= sqrt(xˆ2+yˆ2);

and store the content in a file named (e.g.) mydefs.mac. This file can then loaded into one’sMaxima session and function norm can be used just like any other Maxima function.

(%i1) norm(3,4);

(%o1) norm(3, 4)

(%i2) load("mydefs.mac");

(%o2) myde f s.mac

(%i3) norm(3,4);

(%o3) 5

The line /*...*/ in the above example is ignored when the file is loaded. It is strongly recom-mended to add such comments into one’s files. If files get longer or the coded functions get moresophisticated it helps to maintain one’s code.

ImportantDo not forget the end-of-input marker ; or $ at the end of each piece of code. BImportantUse a string (here "mydefs.mac") as argument for load and do not forget the file extension B(.mac) when you want to load your own file with Maxima code.

2.19 LISP

LISP (from LISt Processing ) is an old programming language and has be developed in 1958 at MIT.It was very popular in the Artificial Intelligence research community.

The core of Maxima is written in LISP. Thus when a expression is entered it is parsed and evaluatedby the underlying LISP interpreter. Fortunately, we do not need any knowledge of this language.However, there are two incidences where you might come in contact with the interpreter. The firstone has already been mentioned in Sect 2.9 (Help) when one types ?sqrt. The second one occursin case of a syntax errors. Usually Maxima will try to explain your error. However, sometimes theLISP interpreter gets confused and responses with a rather obscure error message. Then fix yourinput and ignore the content of this message.

19

2 Basics

— Exercises

1. Store the so-called golden ration, 12 (1 +

√5), in variable gs. In addition compute its numer-

ical value.

2. Verify, that the golden ratio is a root of x2 − x− 1. (Do not use solve!)

3. Compute all roots of x2 − x− 1.

4. Compute (-1)ˆ(1/3). How do you get all (complex) cubic roots of−1?

5. Compute the first 100 digits of π. (Do not forget to reset the precision of big float arith-metic!)

6. Define function ln that computes the natural logarithm.

7. Define a function log10(x) that computes the common logarithm of x. Numerical inputshould be evaluated in floating point. Compute log10(1000).

8. Define a function heron(a,b,c) that computes the area of a triangle with sides a, b, and cby means of Heron’s formula

Area =14

√(a2 + b2 + c2)2 − 2(a4 + b4 + c4) .

9. Get all details about routine solve.

10. Get a list of all Maxima names that contain the (sub-) string solve.

11. Clear your working space. Remove all your variable and function definitions.

20

3 Symbolic Computation

3.1 Symbolic Calculator

In Maxima everything is an expression, including variables, functions, mathematical expressions,objects and programming constructs. Numbers are thus just a special case. Hence Maxima cansolve algebraic equations symbolically in the same way as we do it by hand.

(%i1) solve(a*x^2+b*x+c=0,x);

(%o1) [x = −√

b2 − 4 a c + b2 a

, x =

√b2 − 4 a c− b

2 a]

Of course the coefficients and variables of this quadratic equation can be more complex expres-sions.

(%i2) solve((s-t)*f(y)^2+abs(s+t)*f(y)+(s+t)=0, f(y));

(%o2) [ f (y) =|t + s|+

√5 t2 + 2 s t− 3 s2

2 t− 2 s, f (y) = −

√5 t2 + 2 s t− 3 s2 − |t + s|

2 t− 2 s]

When an expression is evaluated, then Maxima substitutes all expressions to which values areassigned but leaves all other (unknown) expressions left as-is.

(%i3) foo(3*bar+2*bar) + bar^2 + log(%e);

(%o3) f oo(5 bar) + bar2 + 1

ImportantNotice that due to this design of every CAS Maxima cannot detect any typo. If one mistypes a Bvariable or function name, then the invalid name is simply returned as-is.

In Sect. 2.7 we have introduced the assignment operators : and :=. Of course they can be usedto assign any expression to a variable or function. Known variables are then substituted whileunknown expressions are used as-is. wxMaxima tries to print these variables in a nice way. E.g.,names of Greek letters are printed by means of the corresponding symbol.

(%i4) x: (a+beta)*(a-beta);

(%o4) (a− β) (β + a)

(%i5) y: x-a^2;

(%o5) (a− β) (β + a)− a2

(%i6) foo(u):= log(u)-x+alpha;

(%o6) f oo(u) := log(u)− x + α

(%i7) foo(y);

(%o7) log((a− β) (β + a) + 2 a2

)− (a− β) (β + a) + α

21


ImportantWhen an expression has been assigned to some variable, say x, then it remains persistent untilBit is killed (or Maxima is stopped). Now suppose you are running a Maxima session on yourcomputer for hours and use it as a symbolic calculator every now and then, then you might forgetabout such assignments and you might get dubious results or obscure error messages1. Fromthe authors experience this is a very annoying method of “shooting yourself in the foot”. So it isstrongly recommended to kill variable and function definitions that are obsolete.

(%i8) /* ... Do a lot of computations ... *//* Now we want to solve an equation in x and y and define: */eq: a*x+b*y=c;

(%o8) b((a− β) (β + a)− a2

)+ a (a− β) (β + a) = c

(%i9) /* Oops! We expected: */ kill(all)$

(%i10) eq: a*x+b*y=c;

(%o10) b y + a x = c

3.2 Expand, Factor, and Simplify

Command Description

expand(expr) Expand expression exprratexpand(x) Expand rational expression expr (more efficient than expand)factor(expr) Factor expression exprratsimp(expr) Simplifies a (rational) expression exprfullratsimp(expr) Apply ratsimp and similar until no further changes occurradcan(expr) Simplifies expression which contains exponentials, logs, and radicalsnum(expr) Numerator of exprdenom(expr) Denominator of exprtrigsimp(expr) Simplifies expression that contains trigonometric functionstrigreduce(expr) Combines products and powers of trigonometric functions

When Maxima evaluates expressions it usually does not perform all “obvious” expansions or sim-plifications. It also does not try to transform its results into a “nice” form, either. However, Maximaprovides a couple of commands that try to do particular simplifications or transformations of anexpression. Each of these applies several rules which embody conventional notions of simplic-ity. Simplification of expression is quite a difficult job for a computer algebra system even thoughthere are established routines for standard simplification procedures.

Products of sums can be multiplied out by means of expand and ratexpand. To the contrary onecan factor an expression into irreducible terms (which are called prime, e.g., prime numbers).

(%i1) (x+y)*(x-y);

(%o1) (x− y) (x + y)

(%i2) expand(%);

(%o2) x2 − y2

(%i3) factor(%);

(%o3) −(y− x) ∗ (y + x)

1In the worse case you might receive an erroneous results that looks correct at a first glance.

22

3.2 Expand, Factor, and Simplify

(%i4) (x+y)^4;

(%o4) (y + x)4

(%i5) expand(%);

(%o5) y4 + 4 x y3 + 6 x2 y2 + 4 x3 y + x4

(%i6) factor(%);

(%o6) (y + x)4

(%i7) factor(-8*y-4*x+z^2*(2*y+x));

(%o7) (2 y + x) (z− 2) (z + 2)

(%i8) factor(20!);

(%o8) 218 ∗ 28 ∗ 54 ∗ 72 ∗ 11 ∗ 13 ∗ 17 ∗ 19

Numerator of rational terms are also split into their respective terms. Command ratexpand alsocancels out common divisors in rational expressions.

(%i9) e: (x+y)^2/(x^2-y^2);

(%o9)(y + x)2

x2 − y2

(%i10) expand(e);

(%o10)y2

x2 − y2 +2 x y

x2 − y2 +x2

x2 − y2

(%i11) ratexpand(e);

(%o11) − yy− x

− xy− x

Maxima provides commands to find equivalent but simpler expression for more complex one.Thus it applies several rules which embody conventional notions of simplicity. Simplification ofexpression is quite a difficult job for a computer algebra system even though there are establishedroutines for standard simplification procedures.

(%i12) (x^2-y^2)/(x+y)^2;

(%o12)x2 − y2

(y + x)2

(%i13) ratsimp(%);

(%o13) −y− xy + x

(%i14) (%e^x-1)/(1+%e^(x/2));

(%o14)%ex − 1

%ex/2 + 1(%i15) radcan(%);

(%o15) %ex/2 − 1

Commands num and denom allow to decompose a ratio into its numerator and its denominator.

(%i16) num(a/b);

(%o16) a

(%i17) denom(a/b);

(%o17) b

23


(%i18) (3+cos(x)^4+8*sin(x)+4*(cos(x)^2-sin(x)^2)-6*cos(x)^2*sin(x)^2+sin(x)^4)/(8*cos(x)^3);

(%o18)3 + cos(x)4 + 8 sin(x) + 4 (cos(x)2 − sin(x)2)− 6 cos(x)2 sin(x)2 + sin(x)4

8 ∗ cos(x)3

(%i19) trigsimp(%);

(%o19)sin(x) + cos(x)4

cos(x)3

ImportantMaxima offers many more tools to simplify, expand, factor, and contract symbolic expressions.BSystem variables allow a fine-grained adjustment of manipulation. Some of these methods areoffered by wxMaxima’s menubar. However, it is beyond the scope of this tutorial to describe allthe details.

3.3 Verbs and Nouns


'expr Prevents evaluation of expression expr''expr Causes evaluation of expression expr

Maxima distinguishes between nouns and verbs. A verb is an operator which can be executed.A noun is an operator which appears as a symbol in an expression, without being executed. Bydefault, function names are verbs.

The single quote operator ' prevents evaluation of an expression. This can be useful if one needsto “mask” a global variable. Applied to a function call it returns the noun form of the function call.

(%i1) y: 2;

(%o1) 2

(%i2) 'y^2 = y^2;

(%o2) y2 = 4

(%i3) f(x):= x^2;

(%o3) f (x) := x2

(%i4) f(a);

(%o4) a2

(%i5) 'f(a);

(%o5) f (a)

The quote-quote operator '' (two single quote marks) causes evaluation of an expression exprwhen evaluation is otherwise suppressed, such as in function definitions. The value of expr isthen substituted for expr in the input expression. Applied to the operator of an expression, thequote-quote operator changes the operator from a noun to a verb (if it is not already a verb).

(%i6) f(x):= x^2+y^2;

(%o6) f (x) := x2 + y2

(%i7) f(x):= x^2+''y^2;

(%o7) f (y) := x2 + 22

24

3.3 Verbs and Nouns

This is quite convenient when the result of a previous computation should be used as functionbody of a new function. In the following example, we define a function f and its derivative fxwhich we compute by means of command diff2.

(%i8) f(x):= exp(-x^2);

(%o8) f (x) := exp(−x2)

(%i9) fx(x):= ''(diff(f(x),x));

(%o9) f x(x) := −2 x %e−x2

ImportantNotice that the single quote operator ' and the quote-quote operator '' act on the next variable Bor function only. In particular 'f(x) only prevents symbol f from being evaluated but not itsargument x. When the respective operator should prevent or cause evaluation for a more complexexpression, e.g. of f(x), then it must be enclosed in parentheses, '(f(x)).

The following examples demonstrates the behavior. Here only the symbol diff is evaluated (withvalue diff) not the entire expression diff(f(x),x) as in the example above.

(%i10) f(x):= exp(-x^2);

(%o10) f (x) := exp(−x2)

(%i11) fx(x):= ''diff(f(x),x);

(%o11) f x(x) := di f f ( f (x), x)

Here is a simple example for the quote operator.

(%i12) y: 2$

(%i13) f(x):= exp(-x^2)$

(%i14) f(y);

(%o14) %e−4

(%i15) 'f(y);

(%o15) f (2)

(%i16) f('y);

(%o16) %e−y2

(%i17) '(f(y));

(%o17) f (y)

2See Sect. 7.3 for command diff.

25


3.4 Logical Operators


= Equation operatorequal(x,y) Equivalence# Negation of equation operator< Less than<= Less than or equal> Greater than>= Greater than or equaland Conjunction operatoror Disconjunction operatornot Negation operatortrue True statementfalse False statementunknown Statement with unknown logical valueis(x) Attempts to determine whether predicate x is true or false

Operator is attempts to determine the logical value of a predicate. If the predicate is provablythen the respective value true or false is returned. Otherwise, is returns unknown.

The operators and, or, and not force evaluation of its operands, i.e., similar to a call to is.

(%i1) 3>4;

(%o1) 3 > 4

(%i2) is(3>4);

(%o2) f alse

(%i3) not(3>4);

(%o3) true

(%i4) 3>4 or 7<8;

(%o4) true

(%i5) is(x>0);

(%o5) unknown

ImportantOperators = and # test for syntactical equality, in contrast to the equivalence operator equal.BExpressions can be equivalent and not syntactically equal.

(%i6) a: (x+1)*(x-1);

(%o6) (x− 1) (x + 1)

(%i7) b: x^2-1;

(%o7) x2 − 1

(%i8) is(a=b);

(%o8) f alse

(%i9) is(equal(a,b));

(%o9) true

26

3.5 Conditional Evaluation

3.5 Conditional Evaluation


if ... then ... else ... Conditional evaluation

Functions like the absolute value of (real) number are defined by different function terms for dif-ferent subintervals of its domain, e.g.,

abs(x) ={

x, if x ≥ 0,−x, if x < 0.

We can implement such functions using conditional evaluation

if (cond) then (val_1) else (val_0).

There cond is a logical statement composed by means of logical operators (Sect. 3.4). If conditioncode is proved to be true, then the expression evaluates to val_1. Otherwise, if it is false, thenthe expression evaluates to val_0. If is(cond) is unknown, then the whole expression remainsunevaluated.

(%i1) fabs(x):= if (x>=0) then x else -x;

(%o1) i f x >= 0 then x else −x

3.6 Assumptions and Facts


assume(pred) Adds predicate pred to list of “facts”declare(x,prop) Assigns property prop to symbol xfacts() Retrieve list of assumptionsforget(pred) Remove assumption pred

We work with symbols since we are interested in general solutions to our problems. Neverthe-less, we have some knowledge about the variables and parameters in our models. For example,the arguments of a utility function are assumed to be nonnegative real numbers. The parameterof a Cobb-Douglas production function fulfill some inequalities (to ensure concavity). However,when doing symbolic computations Maxima always assumes the general case and thus often doesnot provide a solution (e.g., because there might be infinitely many in the field of complex num-bers). Moreover, for some computations (e.g., when we want to compute the integral

∫xadx) Max-

ima asks about further properties of given symbols (e.g., whether a − 1 is zero or not in∫

xadx),because otherwise Maxima cannot give an closed form solution. For this purpose we can makeassumptions about particular symbols. Thus there is no need for Maxima to ask.

Assumptions can be added as an logical expressions using assume. When Maxima evaluates alogical expression, then it takes care about our assumptions. Command facts retrieves the list ofour assumptions. We can use forget to remove assumptions again.

27


(%i1) is(x>=0);

(%o1) unknown

(%i2) is(U(x)>=0);

(%o2) unknown

(%i3) assume(x>0,U(x)>0);

(%o3) [x > 0, U(x) > 0]

(%i4) is(x>=0 and U(x)>=0);

(%o4) true

(%i5) facts();

(%o5) [x > 0, U(x) > 0]

It is possible to add more assumptions and remove some of them.

(%i6) assume(y>0);

(%o6) [y > 0]

(%i7) facts();

(%o7) [x > 0, U(x) > 0, y > 0]

(%i8) forget(U(x)>0);

(%o8) [U(x) > 0]

(%i9) facts();

(%o9) [x > 0, y > 0]

ImportantWhen we need to assume equality, then we must use equal but not the equality operator =.B

(%i10) assume(a=1);assume: argument cannot be an '=' expression; found a=1assume: maybe you want 'equal'.-- an error. To debug this try: debugmode(true);

(%i11) assume(equal(a,1));

(%o11) [equal(a, 1)]

Unfortunately Maxima is not very clever in deducing the logical value of a statement.

(%i12) assume(x+y<1);

(%o12) [−y− x + 1 > 0]

(%i13) facts();

(%o13) [x > 0, y > 0,−y− x + 1 > 0]

(%i14) is(x*y>0);

(%o14) true

(%i15) is(x<1);

(%o15) unknown

However, Maxima does detect redundant and inconsistent predicates (as long as is(pred) isevaluated to false).

28

3.7 Local Evaluations

(%i16) facts();

(%o16) [x > 0, y > 0,−y− x + 1 > 0]

(%i17) assume(x>0);

(%o17) [redundant]

(%i18) assume(x<0);

(%o18) [inconsistent]

(%i19) facts();

(%o19) [x > 0, y > 0,−y− x + 1 > 0]

In addition to state assumptions Maxima allows to assign properties to symbols. For example wemay declare symbol n to be an integer.

(%i20) declare(n,integer);

(%o20) done

(%i21) facts();

(%o21) [x > 0, y > 0,−y− x + 1 > 0, kind(n, integer)]

3.7 Local Evaluations

Command Description

ev(x,envir) Evaluate expression x with in given environment envir

Maxima uses a couple of system variables for a fine-grained control of the system. E.g., in Sect. 2.11we have seen that the variable fpprec controls the number of significant digits in bigfloat num-bers. Now suppose that we want to compute the first 50 digits of Euler’s number e but not want tomodify fpprec globally. This can be performed by changing its value locally within function ev.

(%i1) ev(bfloat(%e), fpprec:100);

(%o1) 2.7182818284590452353602874713526624977572470937b0

If ev is not used as part of another expression, e.g., in a function, then it is sufficient just to type itsarguments.

(%i2) bfloat(%e), fpprec:100;

(%o2) 2.7182818284590452353602874713526624977572470937b0

29


— Exercises

12. Multiply out (a + b)10.

13. Factor a10 − b10 and a10 + b10.

14. Simplifya10 − b10

a2 − b2 .

15. Find all solutions x of

x2 + 2 (sin(α) + cos(α)) x + sin(2 α) + 1 = 0.

16. Define function sgn(x) the determines the sign of number x. Compute sgn(x) for −2, 0,and π.

sgn(x) =

−1, if x < 0,0, if x = 0,1, if x > 0.

17. Add a minimal set of predicates to your facts such that α > 0, β > 0, α + β < 1, α β > 0,(α− 1) (β− 1) > 0, α (1− α) > 0, β (1− β) > 0, and α (α− 1) < 0, β (β− 1) < 0 evaluate all totrue. What are then the results of is(alpha>-1/2), is(alpha>1/2) and is(alpha>1)?Remove your assumptions when you are done.

18. Define function numeric(x,prec) which evaluates expression x in bigfloat with precisionprec (number significant digits). Use this function and compute the first 30 digits of 1/e.

30

4 Plotting

Plotting graphs of univariate an bivariate functions is an important tool to investigate the proper-ties of such functions. For this task Maxima uses external plotting packages. Thus one can chooseamong two powerful plotting devices1:

• gnuplot (the default)See http://www.gnuplot.info/ for further information.

• xmaxima (formerly called openmath )

Maxima’s plotting functions calculate a set of points and pass them to the plotting package to-gether with a set of commands. All these points and plotting commands are saved into the filemaxout.gnuplot which is located in your home folder.

Within wxMaxima you can easily switch between these plotting packages when the plot menu isused. Notice that both packages, gnuplot and xmaxima open a window that shows the requestedplot. Therefore, wxMaxima offers a third device

• inline

that draws the plots into the notebook rather than a separate window. By right clicking on theinline plot and choosing a destination folder you can save it as an image file.

In this manual we only describe package gnuplot.

The standard commands for plotting in Maxima are plot2d and plot3d. For the inline devicewxMaxima uses special functions wxplot2d and wxplot3d, respectively.

In general there are three types of plots: explicit, implicit, and parametric plots. In fact the syntaxlisted in the sections below are nearly identical.

1. Explicit: The function is given in terms of the input values. For example the functions y =f (x) = x2 defined in R, or z = f (x, y) = sin(x y) defined over the rectangle [−2, 2]× [−3, 2].

2. Implicit: The function is implicitly given by solving an equation of the form F(x, y) = 0. Anexample is x2 + y2 − 1 = 0 in the unit square [0, 1]× [0, 1].

3. Parametric: The plotted curve is given as a path, that is a map (0, 1)→ R2. For example, theparabola y = x2 can be parametrized by using a free parameter named t, and setting x = tand y = t2.

1Do not be afraid. Both packages are installed together with Maxima.

31

4 Plotting

4.1 Plots in 2D

4.1.1 Explicit: Univariate Functions

Command Description

plot2d(funct,x_range,opts) Plot graph of a function of one variable

Option Default Description

[x,x_min,x_max] mandatory Domain x_range of variable x[y,y_min,y_max] automatic Range of vertical axis (“y-axis”)

The most common use of plot2d is to draw the graph of a function. The first argument funct isan expression that describes the function in question. The second argument x_range is manda-tory and is a list [x, x_min, x_max] that contains the name of the variable (usually x but othervariable names are also possible), and the minimum and maximum values, x_min and x_max,respectively. In addition options opts may be given as third, forth, . . . , argument, each of which isa list enclosed in square brackets, see Sect. 4.3.

(%i1) plot2d(sin(x), [x,-%pi,%pi]);

(%o1)

-1

-0.5

0

0.5

1

-3 -2 -1 0 1 2 3

sin(x

)

x

The range for the vertical axis can be adjusted by an optional argument with the form[y, y_min,y_max] (the keyword y is always used for the vertical axis). If that option is used, the plot will showthat exact vertical range, independently of the values reached by the plot. If the vertical range isnot specified, it will be set up according to the minimum and maximum values of the secondcoordinate of the plot points.

32

4.1 Plots in 2D

(%i2) plot2d(sin(x), [x,-%pi,%pi], [y,-2,2]);

(%o2)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

sin(x

)

x

4.1.2 Discrete Points

Command Description

plot2d([discrete,[x_coords,...],[y_coords,...]],opts) Plot pointsplot2d([discrete,[[x_coord,y_coord],...],opts) Plot points


style lines Plotting style

A plot can also be defined in discrete form, i.e., a set of points with given coordinates. A discreteplot is defined by a list starting with the keyword discrete, followed by one or two lists of values.If two lists are given, they must have the same length; the first list will be interpreted as the xcoordinates of the points to be plotted and the second list as the y coordinates. If only one list isgiven after the discrete keyword, each element on the list should also be a list with two values thatcorrespond to the x and y coordinates of a point.

The first example shows the points joined by lines.

(%i3) plot2d([discrete, [10,20,30,40,50], [.6,.9,1.1,1.3,1.4]]);

(%o3)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

10 15 20 25 30 35 40 45 50

y

x

33

4 Plotting

The second example just plots the points. Notice that this time we have given the points in singlelist where each point is a list of two elements.

(%i4) plot2d([discrete, [[10,.6],[20,.9],[30,1.1],[40,1.3],[50,1.4]]],[style,points]);

(%o4)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

10 15 20 25 30 35 40 45 50

y

x

4.1.3 Parametric Plots

Command Description

plot2d([parameter,x_coord,y_coord,t_range],opts) Parametric plot


[t,t_min,t_max] mandatory Domain t_range of “parameter” tnticks 29 Number of points used for plot

A parametric plot is defined by a list starting with the keyword parametric, followed by two ex-pressions which are functions of the “parameter” for each coordinate, and a range for the param-eter. The range for the parameter must be a list with the name of the parameter (usually t butother names are also possible) followed by its minimum and maximum values. The plot will showthe path traced out by the point with coordinates given by the two expressions or functions, asparameter t increases from t_min to t_max.

Option nticks sets the number of points shown in the plot. It controls the smoothness of theparametric curve.

34

4.1 Plots in 2D

(%i5) plot2d([parametric, t*sin(2*%pi*t), t*cos(2*%pi*t), [t,0,2],[nticks, 1000]]);

(%o5)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5

t*co

s(2

*%pi*

t)

t*sin(2*%pi*t)

4.1.4 Combining Plots

Command Description

plot2d([plot_1,...,plot_n],opts) Displays several plots

It is possible to combine two or more plots into one picture. The plots have to be given as a list.Here is an example which combines the graph of f (x) = x3 and a parametric plot of a circle.

(%i6) plot2d([x^3, [parametric,cos(t),sin(t),[t,-%pi,%pi]]],[x,-1,1]);

(%o6)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

x

x^3cos(t), sin(t)

35

4 Plotting

4.1.5 Implicit: Univariate Functions

Command Description

implicit_plot(funct,x_range,y_range,opts) Implicit plot

For plotting implicit functions we must load the package implicit_plot.

The syntax of an implicit plot is a little bit different from command plot2d. It requires that inter-vals for both variables involved in the relation are be specified.

(%i7) load(implicit_plot)$

(%i8) implicit_plot(x^2-y^3+3*y-1=0, [x,-4,4], [y,-4,4]);

(%o8)

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

y

x

4.2 Plots in 3D

4.2.1 Graphs of Bivariate Functions

Command Description

plot3d(funct,x_range,y_range,opts) Plot graph of bivariate function funct

In general the plot commands for plots in three dimensions are similar to the command in twodimensions. We focus on graphs of bivariate functions. The only modification needed is the ad-dition of the second variable. Furthermore, the mouse can be used to rotate the plot looking atthe surface from view points. Just click with your mouse on the image and keep the mouse buttonpressed while moving the mouse. (Unfortunately it depends on your OS which mouse button hasto be used.)

36

4.2 Plots in 3D

(%i1) plot3d(2^(-x^2+y^2),[x,-3,3],[y,-2,2]);

(%o1)

-3 -2 -1 0 1 2 3 -2-1.5

-1-0.5

0 0.5

1 1.5

2

0 2 4 6 8

10 12 14 16

z

2^(y^2-x^2)

x

y

z


Command Description

plot3d([x,y,z],s_range,t_range,opts) Parametric plot in 3d

The next example demonstrates a parametric plot. The surface is given by three expression forthe three coordinates as functions of two “parameters” often called s and t. It shows the famousMöbius strip.

(%i2) moeb_x: cos(s)*(3+t*cos(s/2))$

(%i3) moeb_y: sin(s)*(3+t*cos(s/2))$

(%i4) moeb_z: t*sin(s/2)$

(%i5) plot3d([moeb_x,moeb_y,moeb_z], [s,-4,4], [t,-4,4]);

(%o5)

-4 -2 0 2 4 6 8 -6-4

-2 0

2 4

6

-4-3-2-1 0 1 2 3 4

z

Parametric function

x

y

z

37

4 Plotting

4.2.3 Contour Plots

Command Description

contour_plot(funct,x_range,y_range,opts) Contour plot

A contour plot consists of curves where the given function yields the same value. Unfortunately,one has to pass options to gnuplot to set the number of contour levels. The contour plot in thefollowing example shows 12 levels.

(%i6) contour_plot(u^3+v^2, [u,-4,4], [v,-4,4], [legend,false],[gnuplot_preamble,"set cntrparam levels 12"]);

(%o6)

-4 -3 -2 -1 0 1 2 3 4

u

-4

-3

-2

-1

0

1

2

3

4

v

4.3 Plot Options

Maxima offers a vast set of options creating plots. Thus we only describe the most importantones. Please see Section “Plotting” in the Maxima manual. Options are always given as list withthe option name as its first entry: [opt, value, ...].

4.3.1 Annotation


[xlabel, "text"] "x" Label for x axis[ylabel, "text"] "y" Label for y axis[zlabel, "text"] "z" Label for z axis

With these options one can insert any text for the labels of the two (plot2d) and three axes(plot3d), respectively.


[legend, "text1", "text2", ...] automatic Labels for curves are shown[legend, false] suppress legend

When a more than one curve is shown in plot (two or more graphs or points in plot2d, or forcontour_plot), then these curves are drawn with different lines (usually in different colors). Alegend in the right upper labels these lines. By default, the names of the expressions or functionswill be used, or the words discrete1, discrete2, . . . , for discrete sets of points. Displaying thelegend can be suppressed by means of [legend,false].

38

4.3 Plot Options

4.3.2 Plotting Range


[x, x_min, x_max] mandatory Range for x coordinate[y, y_min, y_max] Range for y coordinate[z, z_min, z_max] optional Range for z coordinate

When a function is plotted using plot2d, then the range for the variable is obligatory. The range ofthe vertical axes is automatically computed, it will be set up according to the minimum and max-imum values of the second coordinate of the plot points. This may sometimes not appropriate.Then the vertical range can be adjusted using option y.

Similarly, when using plot3d the ranges of two variables must be given and the range of the thirdcoordinate (z) is computed automatically or my be adjusted using option z.

Notice that the variables of the functions given to plot2d and plot3d need not be called x andy. Then options x and y can be still used to adjust the plotting range of the first coordinates. Inparticular, the plotting range may be even larger than the given function plot.

4.3.3 Smoothness Parameters


[nticks, integer] 29 Initial number of points in plot2d

When plotting functions with plot2d, we can control the smoothness of the drawn curve by set-ting the number of points by means of option nticks. For explicit plots this is just the initialnumber of plots that will be automatically increased when the given function has great variation.For parametric plots this is the actual number of points that will be shown for the plot. This oftencan result in curve that does not look as smooth as expected. In this case it is recommended toincrease nticks.


[grid, integer, integer] [30,30] Number of grid points in plot3d

The command plot3d evaluates the given function over a given grid. This grid can be specified bysetting the number of points to use in the x and y directions for three-dimensional plotting. Maybe warned that increasing these numbers may result in very long computation times.

4.3.4 Plotting Styles


[style, type_1, type_2, ...] automatic Plotting styles[style, [type_1], [type_2], ...] automatic Plotting styles

Gnuplot selects some style for each function or set of data points such these can be distinguishedin one plot. However, this may always what your want. E.g., you want to join your data points withline segments. Option style allows to specify such styles manually. For each function or data seta style must be given. Each style can be either

lines for line segments,points for isolated points,linespoints for segments and points, ordots for small isolated dots.

39

4 Plotting

The first three styles accept an optional parameter2 (integer) that specifies the line width and pointsize, respectively. Then each style has the be a list: [lines,2] sets the style to lines of width 2.


[color, color_1, color_2,...] automatic Line/point color

In plot2d and implicit_plot we can defines the color (or colors) for the various curves usingoption color. The colors for each curve are taken from the given list in sequential order.

In plot3d, this option defines the colors used for the mesh lines of the surfaces, when no paletteis being used; one side of the surface will have color color_1 and the other color_2 (or the samecolor if there is only one color).

Available colors are: blue, red, green, magenta, black, cyan.


[point_type, type_1, type_2, ...] automatic Point type

When plotting points we can choose among a list of possible point types. The point type for eachdata set are taken from the given list in sequential order.

Available types are: bullet, circle, plus, times, asterisk, box, square, triangle, delta,wedge, nabla, diamond, lozenge.

4.3.5 Meshes and Shading


[mesh_lines_color, color] black Color used for drawing mesh lines[mesh_lines_color, false] Switch off mesh lines[palette, [palette_1], ...] Colors for shading surface[palette, false] Switch of shading of surface

Command plot3D uses two technique to draws a surface: It draws a mesh according to the givengrid bounds. Then the squares of this mesh are filled using a given color palette. Thus the colorsindicate function values (as on any map of a mountain) or it may mimic shading.

The colors of the surface are given by option palette. Each palette is a list with a keyword fol-lowed by four numbers. The keyword specifies which of the three attributes (hue, saturationor value) will be increased according to the values of z. The first three numbers, which must bebetween 0 and 1, define the basis color to be assigned to the minimum value of z. The last numberindicates the increase corresponding to the maximum value of z.Default for the first plot is: [hue, 0.25, 0.7, 0.8, 0.5]

If palette is given the value false, the surfaces will not be shaded but represented with a meshof curves only. In that case, the colors of the lines will be determined by the option color (seeSect. 4.3.4).

Otherwise the color of mesh lines is set my option mesh_lines_color. It accepts the same colorsas for the option color. It can also be given a value false to eliminate completely the mesh lines.

4.3.6 Examples for plot2d

In our first example we plot two functions denoted by first function and second function.Labels for x and y coordinates are abscissa and ordinate, respectively. We use 100 initial points

2In fact there are even more optional parameters, see plot option style in Section Plotting of the manual for more details.

40

4.3 Plot Options

for each plot and restrict the vertical axis to the interval [0, 1.5]. The first function is plotted by athick magenta line while the second function is plotted by a thin blue one.

(%i1) plot2d([exp(-x^2/2), sqrt(%e)*exp(-x)], [x,0,2],[y,0,1.5], [nticks, 100],[legend,"first function", "second function"],[xlabel, "abscissa"], [ylabel, "ordinate"],[style, [lines,10],[lines,1]], [color, magenta,blue]);

(%o1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2

ord

inate

abscissa

first functionsecond function

The next example shows data points together with an approximating function.

(%i2) data_list: [[0,-4], [0.5,-1], [1,0.25], [1.2,0.4], [1.5,0.1],[1.7,0.3], [2,1], [2.2,0.75], [3,1.1]]$

(%i3) plot2d([log(x), [discrete, data_list]], [x,0,5],[legend, "theory", "experiment"],[style, [line,2], [points]],[color, blue, red], [point_type, triangle]);

(%o3)

-7

-6

-5

-4

-3

-2

-1

0

1

2

0 1 2 3 4 5

x

theoryexperiment

4.3.7 Examples for plot3d

In the first example we only use a mesh but do not use shading. We use blue color for the upperside of the surface and red for its lower side. (When we only gave one color, then both sides are

41

4 Plotting

plotted with that color.)

(%i4) plot3d(sin(x*y), [x,-2,2], [y,-2,2],[palette,false], [color, red, blue]);

(%o4)

sin(x*y)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2x -2

-1.5-1

-0.5 0

0.5 1

1.5 2

y-1

-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

z

The next example shows a plot without mesh. In addition the shading is different from the defaultsetting and we use a finer grid.

(%i5) plot3d(sin(x*y), [x,-2,2], [y,-2,2], [grid, 50,50],[mesh_lines_color,false], [palette, [hue, 0.5,0.7,0.6, 0.2]]);

(%o5)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2-1.5

-1-0.5

0 0.5

1 1.5

2

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

z

sin(x*y)

x

y

z

42

4.3 Plot Options

The last example shows the influence of option grid. Here we use a very coarse mesh.

(%i6) plot3d(sin(x*y), [x,-2,2], [y,-2,2], [grid, 5,5]);

(%o6)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2-1.5

-1-0.5

0 0.5

1 1.5

2

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

z

sin(x*y)

x

y

z

— Exercises

19. Plot the graph of the density of the standard normal distribution, f (x) = 1√2π

e−x22 , in the

range [−3, 3]. Label the y axis with density.

20. Plot the graphs of the density of the standard normal distribution, f (x) = 1√2π

e−x22 , and

the Cauchy distribution, g(x) = 1π(1+x2)

, in the range [−3, 3]. Use a thick red line for the

Cauchy distribution, and a thin black line for the normal distribution. The lines should beannotated with Gaussian distribution and Cauchy distribution, respectively. Thevertical axis should be labeled density.

21. Plot the graphs of the power functions xn for n = −2,−1, 0, 1, 2 in the range [0, 2].

22. We are given the set of data points {(.25,−1.03), (.5,−0.63), (.75,−0.28), (1., 0.), (1.25, 0.22),(1.5, 0.38), (1.75, 0.47)}. Plot these points together with function log(x) in the range [0, 2].Choose an appropriate range for the vertical axis.

23. Plot the curve t 7→(

sin(4π t)cos(4π t)

)for t ∈ [0, 4]. What do you see and what do you expect?

24. Plot the graph of the binormal distribution, f (x, y) = 12π exp

(− 1

2 (x2 + y2))

.

(a) Use default setting.

(b) Use a blue mesh, no shading.

(c) Use a mesh that is blue on the upper surface and cyan on the lower one.

(d) Use shading only but no mesh.

43

5 The Draw Package

The abilities of Maxima’s plot command (see Chap. 4) are a bit restricted and do not satisfy allneeds. Moreover, additional parameters for the underlying plotting routine have a syntax that isdifferent from that of Maxima.

Package draw provides a more powerful and flexible plotting (and drawing) tool. In addition morecomplex pictures can be created by means of vector graphics1. However, it only supports the Gnu-plot plotting routine. It thus is a (powerful) Maxima–Gnuplot interface. Before we can use thepackage it must be loaded as described in Sect. 2.17 on p. 18.

(%i1) load(draw)$

It out of the scope of this tutorial to give a detailed description of the package. Thus we refer to theMaxima online help and the corresponding pages in Maxima’s online manual which also includesa link to more elaborate examples.

The layout of this chapters roughly follows that of Chap. 4 (Plotting).

5.1 Draw Scenes

Command Description

gr2d(opts,graphic_objects) Builds an object for a 2D scenegr3d(opts,graphic_objects) Builds an object for a 3D scenedraw(scenes,opts) Plot (a series of) scenesdraw2d(opts,graphic_object) Shortcut for draw(gr2d(opts,graphic_object))draw3d(opts,graphic_object) Shortcut for draw(gr3d(opts,graphic_object))

Roughly spoken the draw package works as follows. Scenes are described in gr2d or gr3d objects,which are then passed to function draw. If more than one scene is described, a multiplot will begenerated. Thus it is possible to construct quite complex pictures. If only one scene is build, thendraw2d(...) and draw3d(...) can be used as these are equivalent to draw(gr2d(...)) anddraw(gr3d(...)), respectively.

Type ?gr2d and ?gr3d to get a list of available graphic objects in two and three dimensions.

Options for these functions (graphics options) are given as key=value pairs.

Scenes in the gr2d and gr3d constructor are interpreted sequentially: graphic options affectthose graphic objects placed on its right. Some graphic options affect the global appearance ofthe scene and may be located at any position or given to the draw command (global options).

1In Vector graphics a figure is composed of elementary geometric objects like lines, polygons, circles, arcs, etc., which arestored by the coordinates of their characteristic points and characteristic parameters like the radius of a circle.

45

5 The Draw Package

5.2 Graphic Objects in 2D

Two dimensional scenes can be build by means of the gr2d scene constructor. There are a coupleof graphic objects available. For plotting graphs of functions we have:

• explicit: The function is given in terms of the input values, e.g., f(x):=xˆ2.

• implicit: The function is implicitly given by the solution of an equation of the form F(x, y) =0, e.g., xˆ2+yˆ2-1=0.

• parametric: The plotted curve is given as a path in R2, e.g., [sin(t*%pi),cos(t*%pi)].

Besides these there are some low level graphic objects for adding lines or annotating graphs. Thefollowing objects have self-explanatory names. For details use Maxima’s online help, e.g., ?points.

• label: Places text on the plot.

• bars

• ellipse

• points

• polygon

• quadrilateral

• rectangle

• triangle

• vector: Draws an arrow.

• region: Plots a region on the plane defined by inequalities.

• image: Plot a bitmap graphic.

In the following we give a short description of the most important graphic objects.

5.2.1 Explicit Functions

Graphic Object Description

explicit(func, x,xmin,xmax) Explicit function with variable x in 2D


xrange=[xmin,xmax] automatic Range of horizontal axis (“y-axis”)yrange=[ymin,ymax] automatic Range of vertical axis (“y-axis”)color black Set color of curve

Plots of explicit functions can be created by means of graphic object explicit. Notice that incontrast to the plot2d command, the domain argument are not given as a list but as argumentsto the explicit call. Of course the argument need not be called x.

46


(%i1) draw2d(explicit(sin(x), x,-%pi,%pi));

(%o1)

-1

-0.5

0

0.5

1

-3 -2 -1 0 1 2 3

The range for the horizontal and vertical axes are computed automatically from the given do-main of the function and the range of the function values. They can be adjusted by the respectiveoptions xrange and yrange. Notice that these are global options and may be specified at anyposition.

The color of the drawn curve can be specified by option color. This is a local option and thus itonly has affect on graphic objects that are placed in its right in the argument list of grd2d. Thusin the following example color has to be set in before adding the explicit function object.

(%i2) draw2d(color=blue, explicit(sin(x), x,-%pi,%pi),xrange=[-4,4], yrange=[-2,2]);

(%o2)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3 4

47

5 The Draw Package

5.2.2 Implicit Functions


implicit(feq, x,xmin,xmax, y,ymin,ymax) Implicit function given by equation feqwith variables x and y in 2D

Implicit functions are defined by an equality relation of the form F(x, y) = 0, e.g., xˆ2-yˆ3+3*y-1=0.It is given similar to the explicit functions but in contrast toexplicit, implicit requires that thedomain for both variables involved have to be specified. The first variable listed in the implicitcall will be plotted on the x-axis and the second variable is plotted on the y-axis. Of course the twoarguments need not be called x and y.

(%i3) draw2d(implicit(x^2-y^3+3*y-1=0, x,-4,4, y,-4,4));

(%o3)

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

y

x



parametric(xfun,yfun, t,tmin,tmax) Parametric function with parameter t in 2D


nticks=integer 29 Number of points used for plot

A parametric plot is defined by a pair of expressions for the x and y coordinates, respectively, anda variable (called parameter) for each of these two functions (often called t but other names arealso possible). The plot will show the path traced out by the point with coordinates given by thetwo expressions as parameter t increases from tmin to tmax.

Option nticks sets the number of points shown in the plot. It controls the smoothness of theparametric curve.

48


(%i4) draw2d(color=blue, nticks=1000,parametric(t*sin(2*%pi*t),t*cos(2*%pi*t), t,0,2));

(%o4)

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1

5.2.4 Discrete Points


points([[x_coord,y_coord],...]) Points in 2Dpoints([x_coords,...],[y_coords,...]) Points in 2D


point_type plus (1) Type of pointpoint_size 1 Size of plotted point.points_joined false If true, then the points are joined by lines.

A plot can also be defined in discrete form, i.e., a set of points with given coordinates. Thesecan be given either by a list of lists of values [x,y], or by two lists of equal length of the x and ycoordinates of each point, respectively.

By default Maxima produces a scatter plot of the points. The style and size of the plotted pointscan be changed by the respective optionspoint_type andpoint_size. Point types can be spec-ified either by a number or the name of the plot style: none (-1), dot (0), plus (1), multiply (2),asterisk (3), square (4), filled_square (5), circle (6), filled_circle (7), up_triangle(8), filled_up_triangle (9), down_triangle (10), filled_down_triangle (11), diamant(12), filled_diamant (13).

49

5 The Draw Package

The first example shows a scatter plot where each point is indicated by a blue filled circle. Thepoints are given as a list of 2-tuples.

(%i5) draw2d(color=blue, point_type=filled_circle, point_size=2,points([[10,.6],[20,.9],[30,1.1],[40,1.3],[50,1.4]]));

(%o5)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

10 15 20 25 30 35 40 45 50

In the second example the points are joined by lines. Moreover, we suppress plotting a mark foreach of the points by using plot style none. The points are given as two lists of the x and y coordi-nates, respectively.

(%i6) draw2d(color=blue, point_type=none, points_joined=true,points([10,20,30,40,50],[.6,.9,1.1,1.3,1.4]));

(%o6)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

10 15 20 25 30 35 40 45 50

50


5.2.5 Combining Plots


key="Text" "" (empty string) Name of function in the legend

It is possible to combine two or more graphic objects in one picture. These are just given as ar-guments to the gr2d or draw2d call. Local graphic options may be given between the graphicobjects. Recall that these local options affect all those graphic objects that are placed on its right. BOption key can be used to add the name of the function or other graphic objects into the legendof the scene. Here is an example which combines the graph of f (x) = x3 (blue) and a parametricplot of a circle (red).

(%i7) draw2d(color=blue,key="x^3", explicit(x^3, x,-1,1),color=red,key="circle", parametric(cos(t), sin(t), t,-%pi,%pi));

(%o7)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

x3

circle


Most of what we have discussed in Sect. 5.2 about 2D graphics also applies here. Three dimen-sional scenes can be build by means of the gr3d scene constructor. Its syntax is similar to that ofthe gr2d. The main difference between these constructors is an additional third variable. Againthere are a couple of available graphic objects. For plotting graphs of functions we have:

• explicit: The function is given in terms of the input values, e.g., f(x,y):=xˆ2+yˆ2.

• implicit: The function is implicitly given by the solution of an equation of the form F(x, y, z) =0, e.g., xˆ2+yˆ2+zˆ2-1=0.

Curves and surfaces in R3 can be produced by the following graphic objects:

• parametric: A parametric curve in R3, e.g., [sin(t*%pi),cos(t*%pi),t].

• parametric_surface: A parametric surface in R3.

Besides these there are some low level graphic objects for adding lines or annotating graphs. Thefollowing objects have self-explanatory names. For details use Maxima’s online help, e.g., ?points.

• label: Places text on the plot.

51

5 The Draw Package

• mesh: Draws a quadrangular mesh in 3D.

• points

• quadrilateral

• triangle

• tube

• vector: Draws an arrow.

There are many, many options available for controlling the appearance of 3D graphics. Althoughwe will describe the most important ones, we refer to the Maxima manual.

Notice that the 3D plots the mouse can be used to rotate the plot looking at the surface from viewpoints. Just click with your mouse on the image and keep the mouse button pressed while movingthe mouse. (Unfortunately it depends on your OS which mouse button has to be used.)

5.3.1 Explicit Functions (Graphs of Bivariate Functions)


explicit(func, x,xmin,xmax, y,ymin,ymax) Explicit function with variables x,y


surface_hide false If true, hidden parts are not plottedenhanced3d none Get colored surfacecolorbox true If true, a color scale is drawn

Similarly to univariate functions we get the graph of a bivariate functions using graphical objectexplicit. Note that in opposition to plot3d only a mesh in drawn.

(%i1) draw3d(explicit(2^(-x^2+y^2), x,-3,3, y,-2,2));

(%o1)

-3 -2 -1 0 1 2 3 -2-1.5-1-0.5 0

0.5 1 1.5 2

0 2 4 6 8

10 12 14 16

52


Option surface_hide controls whether the surface is opaque or not. In the first case hiddenparts are not drawn.

(%i2) draw3d(explicit(2^(-x^2+y^2), x,-3,3, y,-2,2),surface_hide=true);

(%o2)

-3 -2 -1 0 1 2 3 -2-1.5-1-0.5 0

0.5 1 1.5 2

0 2 4 6 8

10 12 14 16

For colored surfaced one has to use option enhanced3d. The simplest way is to use value true.For an advanced usage of this option we refer to the corresponding man page. The color scale canbe suppressed by mean of option colorbox=false.

(%i3) draw3d(enhanced3d=true, explicit(2^(-x^2+y^2), x,-3,3, y,-2,2));

(%o3)

-3 -2 -1 0 1 2 3 -2-1.5-1-0.5 0

0.5 1 1.5 2

0 2 4 6 8

10 12 14 16

0 2 4 6 8 10 12 14 16

53

5 The Draw Package

5.3.2 Implicit Functions


implicit(feq, x,xmin,xmax, y,ymin,ymax, z,zmin,zmax) Implicit function

Implicit functions are defined by an equality relation of the form F(x, y, z) = 0. It is given similarto the explicit functions but in contrast to explicit, implicit requires that the domain forall three variables involved have to be specified. The next example shows the implicit functionx2 + sin(y) = z2 in the domain [−2, 2] × [−2, 2] × [−2, 2]. When enhanced3d is true, then thecolors of the surface depend on the z-values.

(%i4) draw3d(implicit(x^2+sin(y)=z^2, x,-2,2, y,-2,2, z,-2,2));

(%o4)

-2-1.5 -1-0.5 0 0.5 1 1.5 2 -2-1.5-1-0.5 0

0.5 1 1.5 2

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-2-1.5-1-0.5 0 0.5 1 1.5 2



parametric(xfun,yfun,zfun, t,tmin,tmax)parametric_surface(xfun,yfun,zfun, s,smin,smax, t,tmin,tmax)

Graphic object parametric also allows to draw curves in 3D. However, in opposition to the 2Dgraphic object (see 5.2.3 on p. 48) it now requires the additional argument zfun for the 3rd dimen-sion. Here is a simple example that draws a helix.

54


(%i5) draw3d(nticks=500, parametric(sin(t),cos(t),t, t,0,6*%pi));

(%o5)

-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8

10 12 14 16 18

A parametric surface in 3D is defined by functions for the x, y, and z coordinates, respectively.Each is a function in two variables (called parameters). Such a surfaces is represented by graphicobject parametric_surface. The next example shows the famous Möbius strip.

(%i6) moeb_x: cos(s)*(3+t*cos(s/2))$

(%i7) moeb_y: sin(s)*(3+t*cos(s/2))$

(%i8) moeb_z: t*sin(s/2)$

(%i9) draw3d(parametric_surface(moeb_x,moeb_y,moeb_z, s,-%pi,%pi,t,-1,1));

(%o9)

-3 -2 -1 0 1 2 3 -3-2

-1 0 1

2 3

-1-0.5

0 0.5

1

55

5 The Draw Package

5.3.4 Contour Plots


contour none Enable contour plotscontour_levels 5 Number or location of contour levels

Contour curves of a function f (x, y) are lines where f (x, y) = c for some constants c. In the drawpackage, contour plots can be created for explicit functions using graphic option contour.Possible values are:

• none: no contour lines are plotted.

• base: contour lines are projected on the xy plane.

• surface: contour lines are plotted on the surface.

• both: two contour lines are plotted: on the xy plane and on the surface.

• map: contour lines are projected on the xy plane, and the view point is set just in the vertical.

The number of contour lines can be controlled by graphic option contour_levels. It either canbe a single number for the number of levels, or can be given explicitly a list or set of numbers, seeits man page for details.

The following example reproduces the contour plot from Sect. 4.2.3 on p. 38. Notice that we haveto use map for this purpose.

(%i10) draw3d(explicit(x^3+y^2, x,-4,4, y,-4,4),contour=map, contour_levels=12);

(%o10)

60 40 20 0

-20 -40 -60

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

In addition we can also draw contour lines directly on the surface of the function graph as wellon the base (the xy plane), or both. Here is an example for both with function f (x, y) = x2 + y2.

56


(%i11) draw3d(explicit(x^2+y^2, x,-4,4, y,-4,4),contour=both, contour_levels=15);

(%o11)

-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 0 5

10 15 20 25 30

30 25 20 15 10 5

5.3.5 Discrete Points in 3D


points([[x_coord,y_coord,z_coord],...]) Points in 3Dpoints([x_coords,...],[y_coords,...],[z_coords,...]) Points in 3D

It is also possible to plot discrete plots in three dimensions. The commands and options describedin Sect. 5.2.4 on p. 49. In the following example the data points (1, 5, 2), (2, 4, 4), (3, 2, 3), (4, 7, 1),(5, 4, 6) are plotted.

(%i12) xc:[1,2,3,4,5]$

(%i13) yc:[5,4,2,7,4]$

(%i14) zc:[2,4,3,1,6]$

(%i15) draw3d(color=red, point_type=filled_circle, points(xc,yc,zc));

(%o15)

1 1.5 2 2.5 3 3.5 4 4.5 5 2 3

4 5

6 7

1 2 3 4 5 6

57

5 The Draw Package

5.4 Graphic Options

Package draw offers a vast set of options creating and controlling plots. Thus we only describe themost important ones. Please see Section “Draw” in the Maxima manual.

Graphic Options are always given as key=value pairs.

5.4.1 Global Settings

Command Description

set_draw_defaults(opts,...) Set user graphics options

Function set_draw_defaults allows to set global defaults for a Maxima session. Calling thisfunction without arguments removes user defaults again.

The following piece of code changes the default color for plots to red and turns on plotting a gridon the xy plane.

(%i1) set_draw_defaults(color=red, grid=true)$

5.4.2 Annotation and Axes


label(["Text",x_coord,y_coord],...]) Write label(s) in 2Dlabel(["Text",x_coord,y_coord,z_coord],...]) Write label(s) in 3D

One can use graphic object label to write any text anywhere in the plot.


title="Text" "" (empty string) Main title for scenexlabel="Text" "" (empty string) Label for x axisylabel="Text" "" (empty string) Label for y axiszlabel="Text" "" (empty string) Label for z axisxtics auto Tic marks on x axisytics auto Tic marks on y axisztics auto Tic marks on z axis

Option title adds a title for the scene. Options xlabel, xlabel, and xlabel inserts a labelfor the corresponding axes. Options xtics,ytics,and ztics controls the way how tic marks aredrawn. For possible values see the corresponding man pages.


key="Text" "" (empty string) Name of function in the legend

Option key inserts Text to annotate a given function in the legend. For each function plot akey must be given before the graphic object. The entry into the legend for further objects can besuppressed by setting key="".

58

5.4 Graphic Options


xaxis false If true, the x axis is shown drawnyaxis false If true, the y axis is shown drawnzaxis false If true, the z axis is shown drawn (3D only)grid false If true, a grid will be drawn on the xy plane

By default there are no coordinate axes in plots of scenes. These can be drawn by adding optionsxaxis=true, yaxis=true and/or zaxis=true. There are further options that controls the style,width of color of the corresponding axis, see the man pages of these graphic options for details.

5.4.3 Plotting Range


x=[x_min,x_max] auto Range for x coordinatey=[y_min,y_max] auto Range for y coordinatez=[z_min,z_max] auto Range for z coordinate

These options set the range of the plotting region for each direction.

Notice that in case of (explicit or implicit) functions the name of these graphic options are inde-pendent of the particular name of the variables for the function, i.e., x, y, z just refer to the first,second, and third coordinate (direction).

5.4.4 Smoothness Parameters


nticks=integer 29 Initial number of points used for 2D function plots

When plotting explicit or parametric functions, we can control the smoothness of the drawncurve by setting the number of points by means of option nticks. For explicit plots this isjust the initial number of plots that will be automatically increased when the given function hasgreat variation. For parametric plots this is the actual number of points that will be shown forthe plot. This often can result in curve that does not look as smooth as expected. In this case it isrecommended to increase nticks.


xu_grid=integer 30 Number of coordinates of the first variable (3D)yv_grid=integer 30 Number of coordinates of the second variable (3D)

For explicit functions and parametric_surface in 3D a grid of sample points is computed.The number of points in each coordinate can be specified by means of the two options xu_gridand yv_grid. Thus the given function has to be evaluated in xu_grid times yv_grid points.


ip_grid=[integer,integer] [50,50] Number of grid points for the first samplingin implicit plots

ip_grid_in=[integer,integer] [5,5] Number of grid points for the second sam-pling

Plots for implicit graphic objects are created in a two step procedure. In the first step the coarsestructure of the implicit plot is computed. In a second step this structure is refined by means of alocal grids. The smoothness of implicit graphic objects is thus controlled by means of ip_gridand ip_grid_in. It may be necessary to increase the values of ip_grid for very rugged func-tions.

59

5 The Draw Package

5.4.5 Export Graphics


terminal 'screen Selects the terminal for plottingfile_name="file" "maxima_out" Name of file for saving graphical output

Option terminal allows to set the output device for graphics. By default all graphics are drawnon the screen. However, plots also can be exported into some file by setting one of the followingterminals:

• 'pdf

• 'eps

• 'png

• 'gif

• 'svg

The output is then stored in a file set by option file_name. The respective extension is thenautomatically added to the file name.

Here is a simple example where we save the plot of the cosine function in PNG format in filecosine.png. Notice that one must not add extension .png.

(%i2) draw2d(explicit(cos(x),x,-2*%pi,2*%pi),terminal='png, file_name="cosine");

5.4.6 Style and Colors


point_type plus (1) Type of point (see Sect. 5.2.4)point_size 1 Size of plotted point


line_type solid Type of line (solid or dots)line_width 1 Width of plotted line


color blue Color lines, points and labels

Plotting of points and lines can be controlled by various options. Colors can be set by means ofoption color and can given as names or in hexadecimal RGB code. Available colors are (for acomplete list see the man page): gray, blue, red, green, magenta, black, cyan, yellow.


surface_hide false If true, hidden parts are not plotted in 3Denhanced3d none Get colored surface in 3Dpalette color Indicates how to map gray levels onto color components

By default 3D graphical objects are drawn as transparent meshes. If the surface should be opaqueuse option surface_hide=true. For colored surfaced one has to use option enhanced3d toenable colored surfaces and option palette to set a particular color set, see the correspondingman pages for details.

60

5.4 Graphic Options

5.4.7 Examples for draw2d

In our first example we plot two functions denoted by first function and second function.Labels for x and y coordinates are abscissa and ordinate, respectively. We use 100 initial pointsfor each plot and restrict the vertical axis to the interval [0, 1.5]. The first function is plotted by athick magenta line while the second function is plotted by a thin blue one.

(%i3) draw2d(nticks=100, yrange=[0,1.5],xlabel="abscissa", ylabel="ordinate",color=magenta, line_width=10, key="first function",explicit(exp(-x^2/2), x,0,2),color=blue, line_width=1,key="second function",explicit(sqrt(%e)*exp(-x), x,0,2));

(%o3)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2

ord

inate

abscissa

first functionsecond function

The next example shows data points together with an approximating function.

61

5 The Draw Package

(%i4) datalist: [[0,-4], [0.5,-1], [1,0.25], [1.2,0.4], [1.5,0.1],[1.7,0.3], [2,1], [2.2,0.75], [3,1.1]]$

(%i5) draw2d(color=blue, key="theory", explicit(log(x), x,0,5),color=red, point_type=filled_up_triangle, point_size=3,key="experiment", points(datalist));

(%o5)

-4

-3

-2

-1

0

1

0 1 2 3 4 5

theoryexperiment

5.4.8 Examples for draw3d

The following example demonstrates that it us also possible to combine two or more plots in 3D.

(%i6) draw3d(key="Gauss", color=blue,explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),key="Plane", color=red, yv_grid=10,explicit(x+y,x,-5,5,y,-5,5),key="Circle", color=dark_green, line_width=3,parametric(2*sin(t),2*cos(t),1, t,0,2*%pi),surface_hide=true);

(%o6)

GaussPlaneCircle

-4 -2 0 2 4 -4-2

0 2

4

-10-5 0 5

10

62

5.4 Graphic Options

— Exercises

25. Plot the graph of the density of the standard normal distribution, f (x) = 1√2π

e−x22 , in the

range [−3, 3]. Label the y axis with density.

26. Plot the graphs of the density of the standard normal distribution, f (x) = 1√2π

e−x22 , and

the Cauchy distribution, g(x) = 1π(1+x2)

, in the range [−3, 3]. Use a thick red line for the

Cauchy distribution, and a thin black line for the normal distribution. The lines should beannotated with Gaussian distribution and Cauchy distribution, respectively. Thevertical axis should be labeled density.

27. Plot the graphs of the power functions xn for n = −2,−1, 0, 1, 2 in the range [0, 2]. Use differ-ent colors for each function.

28. We are given the set of data points {(.25,−1.03), (.5,−0.63), (.75,−0.28), (1., 0.), (1.25, 0.22),(1.5, 0.38), (1.75, 0.47)}. Plot these points together with function log(x) in the range [0, 2].Choose an appropriate range for the vertical axes.

29. Plot the curve t 7→(

sin(4π t)cos(4π t)

)for t ∈ [0, 4]. What do you see and what do you expect?

30. Plot the graph of the binormal distribution, f (x, y) = 12π exp

(− 1

2 (x2 + y2))

.

63

6 Linear Algebra

6.1 Vectors

Operator Description

u+v Sum of vectors u and vu-v Difference of vectors u and vs*v Multiply vector v with scalar su.v Scalar product of vectors u and v (also called dot product or inner product)

Vectors are implemented in Maxima by means of lists, i.e., their elements enclosed in squarebrackets [...], see Sect. 2.13. Vector addition and multiplication with a scalar are performedby operators +, - and *. The scalar product is computed by the dot operator ..

(%i1) [1,0,1]+[1,2,3];

(%o1) [2, 2, 4]

(%i2) [1,0,1]-[1,2,3];

(%o2) [0,−2,−2]

(%i3) 5*[1,2,3];

(%o3) [5, 10, 15]

(%i4) [1,0,1].[1,2,3];

(%o4) 4

ImportantNotice that * performs a pointwise multiplication. B(%i5) [1,2,3]*[1,2,3];

(%o5) [1, 4, 9]

The k-th element of a vector can be retrieved using [k] as the following example shows.

(%i6) x: [1,2,3,4];

(%o6) [1, 2, 3, 4]

(%i7) x[3];

(%o7) 3

Notice that one can also use [k] as an index for any expression and create a vector with symboliccomponents.

(%i8) y[3];

(%o8) y3

(%i9) z: [z[1],z[2]];

(%o9) [z1, z2]

65

6 Linear Algebra

6.2 Matrices

Command Description

matrix(row_1,...,row_n) Create a rectangular matrix with rows row_1,. . . ,row_n

Operators Description

A+B Sum of matrices A and BA-B Difference of matrices A and Bs*A Multiply matrix A with scalar sA.B Product of matrices A and BA^^n n-th power of matrix A, i.e., A.A.· · ·.AA^^(-1) Inverse of matrix A

Matrices are created by means of command matrix which takes the row vectors of the matrix asits arguments. These must have the same length.

(%i1) A: matrix([1,2],[3,4]);

(%o1)

[1 23 4

](%i2) B: matrix([0,1],[2,3]);

(%o2)

[0 12 3

](%i3) A+B;

(%o3)

[1 35 7

](%i4) 5*A;

(%o4)

[5 1015 20

](%i5) A.B;

(%o5)

[4 78 15

](%i6) A^^2;

(%o6)

[7 1015 22

]

Remind that the inverse of A is given by A−1.

(%i7) A^^(-1);

(%o7)

[−2 1

32 − 1

2

]

ImportantNotice that * and ^ perform a pointwise multiplication and exponentiation, respectively.B(%i8) A*B;

(%o8)

[0 26 12

](%i9) A^2;

(%o9)

[1 49 16

]

66

6.3 Operations on Matrices

The i-th rows of a matrix A can be retrieved by means of index [i]. Element aij can be retrievedor replaced using index [i,j].

(%i10) A: matrix([1,2],[3,4]);

(%o10)

[1 23 4

](%i11) A[2];

(%o11) [3, 4]

(%i12) A[2,1];

(%o12) 3

(%i13) A[2,1]: 10;

(%o13) 10

(%i14) A;

(%o14)

[1 2

10 4

]

ImportantVector are usually printed as row vectors. However, Maxima is quite inconsistent when treating Bvectors. Depending on the given expression it is interpreted as row vector or as column vector.

(%i15) A: matrix([1,2],[3,4]);

(%o15)

[1 23 4

](%i16) x: [1,2];

(%o16) [1, 2]

(%i17) A.x;

(%o17)

[5

11

](%i18) x.A;

(%o18)[7 10

]

6.3 Operations on Matrices

Command Description

diagmatrix(n,x) Diagonal matrix of size n with the diagonal elements xident(n) Identity matrix of order ntranspose(A) Transpose of matrix Ainvert(A) Inverse of matrix A (same as A^^(-1))determinant(A) Determinant of matrix A

(Multiples of) identity matrices can be constructed by means of diagmatrix and ident.

(%i1) ident(2);

(%o1)

[1 00 1

](%i2) diagmatrix(2,3);

(%o2)

[3 00 3

]

67

6 Linear Algebra

(%i3) A: matrix([1,2],[3,4]);

(%o3)

[1 23 4

](%i4) transpose(A);

(%o4)

[1 32 4

](%i5) invert(A);

(%o5)

[−2 1

32 − 1

2

](%i6) determinant(A);

(%o6) −2

6.4 Rank of a Matrix

Command Description

rank(A) Rank of matrix A, i.e., dimension of the column space of Anullity(A) Corank of matrix A, i.e., dimension of the null space of Acolumnspace(A) Basis for the column space of Anullspace(A) Basis for the null space of A

Maxima provides commands for computing image and kernel of a matrix.

(%i1) A: matrix([1,2,3],[4,5,6],[7,8,9]);

(%o1)

1 2 34 5 67 8 9

(%i2) rank(A);

(%o2) 2

(%i3) nullity(A);

(%o3) 1

(%i4) columnspace(A);

(%o4) span

147

,

258

(%i5) nullspace(A);

(%o5) span

−363

6.5 Lists (Revised)

Command Description

length(L) Length of list Lfirst(L) First element of list Lsecond(L) Second element of a list Lrest(L,n) Returns L with its first n elements removedlast(L) Last element of list L

68

6.6 Linear Equations

In Sect. 2.13 we already have learned about lists enclosed in square brackets [...]. But in factlists are the basic building block for LISP and thus for Maxima. Almost all data types, includingfunctions are lists. So we can extract any elements from this list.

(%i1) first(f(x,y,z));

(%o1) x

(%i2) rest(f(x,y,z));

(%o2) f (y, z)

(%i3) last(f(x,y,z));

(%o3) z

Commands nullspace and columnspace return the bases of the corresponding spaces. Theelements of these bases can be extracted using list operations.

(%i4) A: matrix([1,2,3],[4,5,6],[7,8,9]);

(%o4)

1 2 34 5 67 8 9

(%i5) CS: columnspace(A);

(%o5) span

147

,

258

(%i6) first(CS);

(%o6)

147

(%i7) second(CS);

(%o7)

258

6.6 Linear Equations

A system of linear equations is a collections of linear equations involving the same set of variables.A general system of m equations with n unknowns can be written as

a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2

......

. . ....

...am1x1 + am2x2 + · · ·+ amnxn = bm

with x1, . . . , xn are the unkowns, a11, . . . , amn are the coefficients of the linear equalities and b1, . . . , bmare the constant on the right-hand side. This notations is equivalent to the matrix equation of theform

A x = bwhere A is called the coefficient matrix of this system of linear equations, vector x contains theunknowns, and vector b contains all the constants:

A =

a11 a12 · · · a1nxna21 a22 · · · a2nxn

......

. . ....

am1 am2 · · · amnxn

, x =

x1x2...

xn

, b =

b1b2...

bm

.

69

6 Linear Algebra

Finding solutions of such linear equations is an importing part for many tools. For a quadratic (i.e.n× n) regular coefficient matrix A one may use its inverse A−1: x = A−1b. For this purpose onecan apply A^^(-1) (see Sect. 6.2) or invert(A) (see Sect. 6.3).

(%i1) A: matrix([1,2,2],[3,3,8],[0,7,4]);

(%o1)

1 2 23 3 80 7 4

(%i2) b: [-1,-15,2];

(%o2) [−1,−15, 2]

(%i3) A^^(-1).b;

(%o3)[1 2 −3

]Notice that there exist other more efficient methods for solving such linear equations. See theMaxima manual for such functions.

Command solve offers a general method that works for any number of equations an unknown,see also Sect. 2.15.

(%i4) eq1: x[1]+2*x[2]+2*x[3]=-1;

(%o4) 2x3 + 2x2 + x1 = −1

(%i5) eq2: 3*x[1]+3*x[2]+8*x[3]=-15;

(%o5) 8x3 + 3x2 + 3x1 = −15

(%i6) eq3: 7*x[2]+4*x[3]=2;

(%o6) 4x3 + 7x2 = 2

(%i7) solve([eq1,eq2,eq3], [x[1],x[2],x[3]]);

(%o7) [[x1 = 1, x2 = 2, x3 = −3]]

solve also can handle under-determined systems of equations as the following example shows.The symbol %r1 is introduced to represent arbitrary parameters in the solution. A particular solu-tion can be found by replacing %r1 by some number.

(%i8) solve([eq1,eq2], [x[1],x[2],x[3]]);

(%o8) [[x1 = −10 %r1 + 273

, x2 =2 %r1 + 12

3, x3 = %r1]]

(%i9) %, %r1=2;

(%o9) [[x1 = −473

, x2 =163

, x3 = 2]]

Notice that the set of all solutions of an under-determined systems of equations forms an affinespace which can be described by a particular solution of Ax = b and the nullspace of A (which isthe set of solutions of the homogeneous equation Ax = 0). Its basis can be obtained by commandnullspace, see Sect. 6.4.

(%i10) nullspace(matrix([1,2,2],[3,3,8]));

(%o10) span

10−2−3

70

6.7 Eigenvalues and Eigenvectors

For the sake of completeness we also remark that one can compute the echelon form of a givenmatrix.

(%i11) echelon(A);

(%o11)

1 2 20 1 − 2

30 0 1


Command Description

charpoly(A,x) Characteristic polynomial of Aeigenvalues(A) Eigenvalues of matrix Aeigenvectors(A) Eigenvectors of matrix A

Command charpoly(A,x) returns the characteristic polynomial for matrix A with respect to vari-able x.

(%i1) A: matrix([1,2,1],[1,4,4],[7,3,5]);

(%o1)

1 2 11 4 47 3 5

(%i2) charpoly(A,x);

(%o2) −7(4− x) + ((4− x)(5− x)− 12)(1− x)− 2(−x− 23) + 3

Eigenvalues can be computed by means of command eigenvalues. It returns a list of two ele-ments. The first one is the list of eigenvalues while the second element is a list of their correspond-ing algebraic multiplicities.

(%i3) B: matrix([1,0],[6,6]);

(%o3)

(1 06 6

)(%i4) eigenvalues(B);

(%o4) [[1, 6], [1, 1]]

(%i5) eigvals: eigenvalues(B)[1];

(%o5) [1, 6]

(%i6) multiplicities: eigenvalues(B)[2];

(%o6) [1, 1]

Command eigenvectors computes both eigenvalues and the eigenvector. It returns a list oftwo elements: the result of eigenvalues and a list of the corresponding eigenspaces. Eacheigenspace is represented by a list that contain its basis vectors (“eigenvectors”).

71

6 Linear Algebra

(%i7) es: eigenvectors(B);

(%o7) [[[1, 6], [1, 1]], [[[1,−65]], [[0, 1]]]]

(%i8) eigvals: es[1][1];

(%o8) [1, 6]

(%i9) multiplicities: es[1][2];

(%o9) [1, 1]

(%i10) vectors: es[2];

(%o10) [[[1,−65]], [[0, 1]]]

Notice that multiple eigenvalues are listed only once as the following example shows. In follow-ing example the algebraic multiplicity of the unique eigenvalue 3 is 2 while the dimension of thecorresponding eigenspace (i.e., its geometric multiplicity) is only 1.

(%i11) C: matrix([3,1],[0,3]);

(%o11)

(3 10 3

)(%i12) eigenvalues(C);

(%o12) [[3], [2]]

(%i13) eigenvectors(C);

(%o13) [[[3], [2]], [[[1, 0]]]]

In the next example algebraic and geometry multiplicities coincide

(%i14) D: matrix([3,0],[0,3]);

(%o14)

(3 00 3

)(%i15) eigenvectors(D);

(%o15) [[[3], [2]], [[[1, 0], [0, 1]]]]

ImportantCommand eigenvalue simply computes the roots of the characteristic polynomial. In the caseBof symmetric matrices we known that all eigenvalues are real. However, the result may look likecomplex numbers. These the imaginary numbers can be removed as described in Sect. 2.16.

72


— Exercises

31. Let x =

123

and y =

456

. Compute x + y, x− y, x′y, and 3x.

32. Let A =

1 2 34 5 67 8 9

and A =

−1 0 2−2 1 32 4 −1

.

Compute A + B, A − B, 3A, A B, A2, B−1, |A|, and A′. What happens when one tries tocompute A−1?

33. Let A =

1 2 34 5 67 8 9

and A =

−1 0 2−2 1 32 4 −1

.

Compute rank and corank of matrices A and B.Compute null space and columns space of A and B.Compute the linear combination α1v1 + α2v2 of the base vectors v1 and v2 of the columnspace of A.

34. Solve the following systems of equations:

(a)2 x1 + 3 x2 + 4 x3 = 2

4 x1 + 3 x2 + x3 = 10x1 + 2 x2 + 4 x3 = 5

(b)2 x1 + 2 x2 + x3 + 3 x4 = 103 x1 + 5 x2 + 2 x3 − x4 = 30

x1 + 2 x2 + x3 − x4 = 12

(c)

2 x1 + 10 x2 + 4 x3 + 9 x4 = 1x1 + 6 x2 + 5 x3 + 3 x4 = 1

3 x1 + 16 x2 + 9 x3 + 11 x4 = −1x1 + 5 x2 + 2 x3 + 5 x4 = 2

x2 + 3 x3 = 4

35. Compute all eigenvalues and eigenvectors of A =

(1 23 4

). Store the list of eigenvalues in

eigval. Store the eigenvectors to the first and second eigenvalue in v[1] and v[2], repec-tively.


1 2 12 1 31 3 1

numerically.

Notice that the matrix is symmetric (and thus all eigenvalues are real).


1 0 00 1 00 0 1

and B =

1 1 10 1 10 0 1

.

What are the algebraic and geometric multiplicities of the corresponding eigenvalues?

73

7 Calculus

7.1 Limits

Command Description

limit(expr,x,val) Limit of expression expr when variable x approaches vallimit(expr,x,val,dir) Limit of expr when x approaches val from direction dir


infinity Complex infinityind Bounded indefinite resultund Undefined result

Command limit computes the limit of a given expression. It requires an expression, name of thevariable with the limit point to approach. It also computes limits to inf (∞) and minf (−∞), re-spectively. When computing the limit Maxima assumes that x is real variable. The output can alsobe one of ∞,−∞, infinity (complex infinity), ind (indefinite but bounded), or und (undefined).Notice that infinity is returned when the limit of the absolute value of the expression is positiveinfinity, but the limit of the expression itself is neither ∞ nor−∞.

(%i1) limit(sin(x)/x,x,0);

(%o1) 1

(%i2) limit(atan(x),x,minf);

(%o2) −π

2(%i3) limit(log(x),x,0);

(%o3) in f inity

(%i4) limit(sin(1/x),x,0);

(%o4) ind

(%i5) limit(sin(1/x)/x,x,0);

(%o5) und

ImportantDo not mix up result infinity with ∞. The latter is represented by inf in Maxima. BIt is also possible to compute limits from above and below by adding a forth argument. It may havethe respective values plus for limits from above and minus for limits from below.

(%i6) limit(log(x),x,0,plus);

(%o6) −∞

(%i7) limit(x/abs(x),x,0,plus);

(%o7) 1

(%i8) limit(x/abs(x),x,0,minus);

(%o8) −1

(%i9) limit(x/abs(x),x,0);

(%o9) und

75

7 Calculus

7.2 Sums and Products

Command Description

sum(expr,i,i_0,i_1) Sum of the values of expr with index iproduct(expr,i,i_0,i_1) Product of the values of expr with index i


simpsum false When true, simplification rules for sums are appliedsimpproduct false When true, simplification rules for products are applied

Command sum evaluates the sum of the first argument expr where index variable i is substitutedfor all values i_0,...,i_i.

(%i1) sum(z(i),i,1,5);

(%o1) z(5) + z(4) + z(3) + z(2) + z(1)

(%i2) sum(z[i],i,1,5);

(%o2) z5 + z4 + z3 + z2 + z1

(%i3) sum(i^3,i,1,10);

(%o3) 3025

Notice that i_1 ≥ i_0 must hold. Otherwise 0 is returned. Upper and lower bound my also beexpressions including minf and inf.

(%i4) sum(z[i],i,1,n);

(%o4)n

∑i=1

zi

(%i5) sum(z[i],i,1,inf);

(%o5)∞

∑i=1

zi

Maxima tries to compute (simplify) sums when the system variable simpsum is true.

(%i6) sum(1/4^i,i,1,inf);

(%o6)∞

∑i=1

14i

(%i7) sum(1/4^i,i,1,inf), simpsum;

(%o7)13

(%i8) sum(1/i^2,i,1,inf);

(%o8)∞

∑i=1

1i2

(%i9) sum(1/i^2,i,1,inf), simpsum;

(%o9)π2

6

76

7.3 Derivatives and Total Differentials

Analogously one can compute products by means of command product.

(%i10) product(i^3,i,1,10);

(%o10) 47784725839872000000

(%i11) product(z(i),i,1,minf);

(%o11)∞

∏i=1

z(i)

(%i12) product(x+i*(i+1)/2,i,1,4);

(%o12) (x + 1)(x + 3)(x + 6)(x + 10)

7.3 Derivatives and Total Differentials

Command Description

diff(expr,x) Derivative of expr with respect to xdiff(expr,x,n) n-th derivative of expr with respect to xdiff(expr,x_1,n_1,x_2,n_2,...) Mixed partial derivativesdiff(expr) Total differential of exprdel(x) Represents differential of x

Maxima is capable of computing derivatives. It provides a single command diff for this purpose.diff(expr,x) differentiates the expression expr with respect to variable x. Optionally, a thirdargument may be given for derivatives of higher order. For example expr,x,2 yields the secondderivative of expr with respect to variable x.

(%i1) diff(x^a,x);

(%o1) a xa−1

(%i2) diff(x^a,x,2);

(%o2) (a− 1) a xa−2

Mixed partial derivatives can be computed either by nesting diff calls or listing all variables asarguments.

(%i3) diff(diff(x^3*y^2,x),y);

(%o3) 6 x2 y

(%i4) diff(x^3*y^2,x,1,y,1);

(%o4) 6 x2 y

(%i5) diff(x^3*y^2,x,2,y,1);

(%o5) 12 x y

ImportantNotice that for mixed partial derivatives each variable name must be immediately followed by the Border of the respective derivative.

When we call diff with only one argument (i.e., without passing any variable names), then Max-ima computes the total differential of the given expression. The differentials of the variables arerepresented by symbol del.

(%i6) diff(sin(y*x));

(%o6) x cos(xy) del(y) + y cos(xy) del(x)

77

7 Calculus

If we prefer dx and dy instead, we have to substitute accordingly as shown in the next example.

(%i7) diff(sin(y*x));

(%o7) x cos(xy) del(y) + y cos(xy) del(x)

(%i8) %, del(x)=dx, del(y)=dy;

(%o8) dx y cos(x y) + dy x cos(x y)

The quote operator ' may be used to prevent Maxima from immediately compute the derivative.

(%i9) 'diff(exp(-x^2),x)=diff(exp(-x^2),x);

(%o9)d

dx%e−x2

= −2 x %e−x2

7.4 Dependencies and Chain Rule

Command Description

depends(f,x) Declare functional dependency of f from x


derivabbrev false When true, symbolic derivatives are displayed as subscriptsdependencies [] List of functional dependencies

We can also work with symbolic derivatives, i.e., the derivatives some function f . However, anydependencies of f must be represented explicitly.

(%i1) diff(f(x,y),x);

(%o1)d

dxf (x, y)

(%i2) diff(f,x);

(%o2) 0

However, Maxima allows to declare dependencies among variables and diff takes fully care ofthis.

(%i3) depends(f,x);

(%o3) [ f (x)]

(%i4) diff(f,x);

(%o4)d

dxf

By default derivatives are displayed in Leibniz notation d fdx (which should read ∂ f

∂x in case of partialderivatives). However, when we set the system variable derivabbrev to true, then symbolicderivatives are displayed as subscript as in fx.

(%i5) derivabbrev: true$

(%i6) diff(f,x);

(%o6) fx

(%i7) derivabbrev: false$

78

7.4 Dependencies and Chain Rule

Now suppose that x itself depends on some other variable t. The we may compute the derivativeof the compound function f (x(t)) with respect to t by means of the chain rule:

d f (x(t))dt

=d

dxf (x(t))

ddt

x(t)

We try the following:

(%i8) diff(f(x(t)),t);

(%o8)ddt

f (x(t))

However, this does not work. The solution is to declare a dependency of x from t (notice thatalready declared f being depend from t).

(%i9) depends(x,t);

(%o9) [x(t)]

(%i10) diff(f,t);

(%o10)

(d

dxf) (

ddt

x)

Do not forget to remove the dependencies when you do not need it any more. Noticedependenciescontains a list of all dependencies declared so far. They can be removed using kill

(%i11) dependencies;

(%o11) [ f (x), x(t)]

(%i12) kill(f)$

(%i13) dependencies;

(%o13) [x(t)]

Here is a more advanced function. Assume we need the derivatives of f (x(t), t) with respect to xand t.

(%i14) depends(f,[x,t]);

(%o14) [ f (x, t)]

(%i15) depends(x,t);

(%o15) [x(t)]

(%i16) diff(f,x);

(%o16)d

dxf

(%i17) diff(f,t);

(%o17)

(d

dxf) (

ddt

x)+

ddt

f

(%i18) kill(all)$

When computing total derivatives Maxima treats every unknown symbol as variable. However, wemay declare some of these variables as constants using command declare, see Sect 3.6.

79

7 Calculus

(%i19) diff(a*x^2+b*y^2);

(%o19) 2 b y del(y) + 2 a x del(x) + y2 del(b) + x2 del(a)

(%i20) declare([a,b],constant);

(%o20) done

(%i21) diff(a*x^2+b*y^2);

(%o21) 2 b y del(y) + 2 a x del(x)

7.5 Jacobian Matrix and Gradient

Command Description

jacobian(f,x) Jacobian matrix of vector-valued function f with respect to vector x

In Maxima Vector-values functions, i.e., functions from Rn → Rm, are represented by vectors of(single-values) functions.

(%i1) f(x,y):= [x^2+y^2, x^2-y^2];

(%o1) f (x, y) := [x2 + y2, x2 − y2]

The derivative of such functions is called the Jacobian matrix.

(%i2) jacobian(f(x,y),[x,y]);

(%o2)

[2 x 2 y2 x −2 y

]

Here is a more symbolic example.

(%i3) f(x,y):= [f[1](x,y),f[2](x,y)];

(%o3) f (x, y) := [ f1(x, y), f2(x, y)]

(%i4) jacobian(f(x,y),[x,y]);

(%o4)

[d

dx f1(x, y) ddy f1(x, y)

ddx f2(x, y) d

dy f2(x, y)

]

The gradient of a single-valued function can be computed as Jacobian. Notice that the functionmust be given as a list with one element. The result is a 1× n-matrix.

(%i5) jacobian([x^3+y^2],[x,y]);

(%o5) [3 x2 2 y]

80

7.6 Taylor Series

7.6 Taylor Series

Command Description

taylor(expr,x,a,n) Expands expr in a Taylor series of order n in variable x around apowerseries(expr,x,a) General form of the power series expansionniceindices(expr) Rename indices of sums and products in expr

A Taylor series is a representation of a function as an infinite sum of terms calculated from thevalues of its derivatives at a single point a.

f (x) =∞

∑k=0

dk fdxk (a) (x− a)k

The n-th order truncated taylor series (also known as the n-th Taylor polynomials) consists of thefirst terms up to degree n.

f (x) ≈n

∑k=0

dk fdxk (a) (x− a)k = f (a) + f ′(a) (x− a) +

f ′′(a)2!

(x− a)2 + · · ·+ f (n)(a)n!

(x− a)n

It can be easily computed by means of command taylor. In the following example we computethe truncated Taylor series of order 3 of ex with expansion point 0.

(%i1) taylor(exp(x),x,0,3);

(%o1) /T/ 1 + x +x2

2+

x3

6+ . . .

The leading symbol /T/ indicates a Taylor series. The trailing ellipses ... represents truncation.The resulting expression can be used in further calculations. For this purpose the truncated Tay-lor series is first transformed into a polynomial that is represented by a rational expression. It isindicated by symbol /R/.

(%i2) T(x):= ''(taylor(exp(x),x,0,3));

(%o2) /T/ T(x) := 1 + x +x2

2+

x3

6+ . . .

(%i3) T(3);

(%o3) /R/ 13

(%i4) T(y);

(%o4) /R/y3 + 3 y2 + 6 y + 6

6

Taylor series of functions in two or more variables are obtained by using vectors for x and point a.Here is a simple example.

(%i5) taylor(exp(x^2+y),[x,y],[0,1],2);

(%o5) /T/ %e + %e (y− 1) +2%e x2 + %e (y− 1)2

2+ . . .

Maxima can also handle infinite Taylor series. Command powerseries creates the infinite sum.

81

7 Calculus

(%i6) powerseries(sin(x),x,0);

(%o6)∞

∑i1=0

(−1)i1x2i1+1

(2i1 + 1)!

(%i7) niceindices(%);

(%o7)∞

∑i=0

(−1)ix2i+1

(2i + 1)!

Maxima automatically uses the indices i1, i2, . . . . Command niceindices replaces these to themore familiar symbols i, j, . . . .

7.7 Minima and Maxima of Univariate Functions

Extrema of a univariate function f can be found by the following well-known method:

1. Find the critical points of f , i.e., points x∗ with f ′(x∗) = 0.

2. Compute the second derivative f ′′ and check its sign at these critical points.

– If f ′′(x∗) > 0, then x∗ is a local minimum.

– If f ′′(x∗) < 0, then x∗ is a local maximum.

– If f ′′(x∗) = 0, then we need higher order derivatives at x∗ for a decision.

Assume we have function f (x) = x4 − 3 x2 + 2 and need to compute all local extrema. It is alwaysa good idea to plot the graph of the function.

(%i1) f(x):= x^4-3*x^2+2;

(%o1) f (x) := x4 − 3 x2 + 2

(%i2) plot2d(f(x), [x,-2,2], [y,-1,3]);

(%o2)

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x^

4-3

*x^

2+

2

x

In order to find all critical points we have to compute the first derivative f ′ and find all its roots.

(%i3) fx: diff(f(x),x);

(%o3) 4 x3 − 6 x

(%i4) crit: solve(fx=0,x);

(%o4) [x = −√

3√2

, x =

√3√2

, x = 0]

82

7.8 Minima and Maxima of Multivariate Functions

Next we have to evaluate the second derivative f ′′ at all these critical points and check the signs ofthe results.

(%i5) fxx: diff(f(x),x,2);

(%o5) 12 x2 − 6

(%i6) fxx, crit[1];

(%o6) 12

(%i7) fxx, crit[2];

(%o7) 12

(%i8) fxx, crit[3];

(%o8) −3

Consequently, we have two local minima at x1 = −√

3√2

and x2 =√

3√2

, and a local maximum atx3 = 0.


Command Description

hessian(f,x) Hessian matrix of function f with respect to vector x

Extrema of a multivariate function f can be found by an analogous recipe:

1. Find the critical points of f , i.e., points x∗ with∇ f (x∗) = 0.

2. Compute the Hessian matrix f ′′ and check its definiteness at these critical points.

– If f ′′(x∗) is positive definite, then x∗ is a local minimum.

– If f ′′(x∗) is negative definite, then x∗ is a local maximum.

– If determinant | f ′′(x∗)| = 0, then we need higher order derivatives at x∗ for a decision.

– Otherwise, f ′′(x∗) is indefinite and x∗ is a saddle point.

The Hessian matrix of a function f consists of all second partial derivatives.

f ′′(x1, . . . , xn) =

∂2 f∂x2

1

∂2 f∂x1∂x2

. . . ∂2 f∂x1∂xn

∂2 f∂x2∂x1

∂2 f∂x2

2. . . ∂2 f

∂x2∂xn

......

. . ....

∂2 f∂xn∂x1

∂2 f∂xn∂x2

. . . ∂2 f∂x2

n

Command hessian computes the Hessian matrix for a given function.

(%i1) hessian(x^3+4*x*y+y^2,[x,y]);

(%o1)

[6 x 44 2

]

83

7 Calculus

Here is an example with symbolic derivatives.

(%i2) depends(g,[x,y])$

(%i3) hessian(g,[x,y]);

(%o3)

d2

dx2 g d2

dx dy gd2

dx dy g d2

dy2 g

Computing local extrema is much more difficult for functions with two or more variables than forunivariate functions. So consider the following function.

(%i4) f(x,y):=1/6*x^3 - x + 1/4*x*y^2;

(%o4) f (x, y) :=16

x3 − x +14

x y2

Again it is a good idea to plot the graph of the function. (Use the mouse to rotate the graph and getviews from different viewpoints.)

(%i5) plot3d(f(x,y), [x,-2,2], [y,-2.5,2.5]);

(%o5)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1 0

1 2

-2.5-2

-1.5-1

-0.5 0

0.5 1

1.5 2

2.5

z

x*y^2/4+x^3/6-x

x

y

z

(%i6) contour_plot(f(x,y), [x,-2,2], [y,-2.5,2.5], [legend,false],[gnuplot_preamble, "set cntrparam levels 12"]);

(%o6)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x

-2

-1

0

1

2

y

84


First we have to find the critical points of f . Thus we compute the gradient of f and have to solvea system of equations.

(%i7) [fx,fy]: [diff(f(x,y),x), diff(f(x,y),y)];

(%o7) [y2

4+

x2

2− 1,

x y2]

(%i8) crit: solve([fx=0,fy=0],[x,y]);

(%o8) [[x = −√

2, y = 0], [x =√

2, y = 0], [x = 0, y = 2], [x = 0, y = −2]]

Next we need the Hessian matrix of f .

(%i9) hess: hessian(f(x,y),[x,y]);

(%o9)

[x y

2y2

x2

]

Remind there are two method for estimating the definiteness of a symmetric matrix A. The firstlooks at the eigenvalues of the matrix:

• A is positive definite⇔ all eigenvalues of A are positive.

• A is negative definite⇔ all eigenvalues of A are negative.

• A is indefinite⇔ A has positive and negative eigenvalues.

(%i10) H1: ev(hess, crit[1]);

(%o10)

[−√

2 00 − 1√

2

](%i11) eigenvalues(H1);

(%o11) [[− 1√2

,−√

2], [1, 1]]


(%o12)

[0 11 0

](%i13) eigenvalues(H3);

(%o13) [[−1, 1], [1, 1]]

Hence x1 = (−√

2, 0)′ is a local maximum and x3 = (0, 2)′ is a saddle point.

The second method makes use of the leading principle minors of a symmetric matrix A. The k-thleading principle minor of A is the determinant of the left upper k× k submatrix.

Mk =

∣∣∣∣∣∣∣∣∣a11 a12 . . . a1ka21 a22 . . . a2k

......

. . ....

ak1 ak2 . . . akk

∣∣∣∣∣∣∣∣∣We then have

• A is positive definite⇔ Mk > 0 for all k = 1, . . . n.

• A is negative definite⇔ (−1)k Mk > 0 for all k = 1, . . . n.

• A is indefinite⇔ Mn = |A| 6= 0 but none of the above cases holds.

Notice that for a 2× 2 matrix this reduces to

85

7 Calculus

• A is positive definite⇔ a11 > 0 and |A| > 0.

• A is negative definite⇔ a11 < 0 and |A| > 0.

• A is indefinite⇔ |A| < 0.


(%o14)

[√2 0

0 1√2

](%i15) [H2[1,1], determinant(H2)];

(%o15) [√

2, 1]


(%o16)

[0 −1−1 0

](%i17) [H4[1,1], determinant(H4)];

(%o17) [0,−1]

Hence x2 = (√

2, 0)′ is a local minimum and x4 = (0,−2)′ is a saddle point.

7.9 Integration

Command Description

integrate(expr,x) Indefinite integral of expr with respect of xintegrate(expr,x,a,b) Definite integral of expr with respect of x with limits a and bquad_qags(expr,x,a,b) Numerical approximation of integral over a finite interval

Command integrate attempts to symbolically compute the integral of a given expression. Thusa range of heuristics is employed to handle cases where closed form solutions are known. If oneof these methods succeeds, then the integral is returned. Notice that integrate usually does notadd an integration constant.

(%i1) integrate(sin(x)^3, x);

(%o1)cos(x)3

3− cos(x)

(%i2) integrate(x/sqrt(b^2-x^2),x);

(%o2) −√

b2 − x2

(%i3) integrate(exp(-x^2),x);

(%o3)

√π er f (x)

2

The last example shows that the integral may contain functions names that are less common. In

this case the so called error function is defined as erf(x) =2√π

∫ x

0e−t2

dt . It plays an important

rôle in statistics. However, there might be even more obscure function that are not used in eco-nomics.

If integrate does not succeed, then the return value is the noun form of the integral or an ex-pression containing one or more noun forms. The noun form is displayed with an integral sign.

(%i4) integrate(tan(log(x)),x);

(%o4)∫

tan(log(x))dx

86

7.9 Integration

integrate may need to know some property of a parameter in the integrand. Thus it will firstconsult the assume database (see Sect. 3.6), and, if the variable of interest is not there, integratewill ask the user. Depending on the question, suitable responses are yes; or no;, or pos;, zero;,or neg;.

(%i5) integrate(x^a,x);Is a+1 zero or nonzero? zero;

(%o5) log(x)

We can avoid this question by adding the answer into ourfacts. Notice that we have to useequalrather then the symbol =. (Do not forget to forget of kill this assumption when it is not neededany more!)

(%i6) assume(equal(a,-1));

(%o6) [equal(a,−1)]

(%i7) integrate(x^a,x);

(%o7) log(x)

Command integrate also handles definite integrals. However, internally it is using an algorithmthat works independently from that for indefinite integrals. Notice that the integral boundariesalso may be minf and inf.

(%i8) integrate(cos(x)^2*exp(x),x,0,%pi);

(%o8)3%eπ

5− 3

5(%i9) integrate(x^2*exp(-x^2),x,minf,inf);

(%o9)

√π

2

The QUADPACK package provides a couple of functions for numerical approximations of definiteintegrals. We give a simple example with routine quad_qags. It returns a list of four elements:

• an approximation to the integral,

• the estimated absolute error of the approximation,

• the number integrand evaluations,

• an error code which is 0 if no problems were encountered.

(%i10) quad_qags(x^(1/2)*log(1/x),x,0,1);

(%o10) [.4444444444444448, 1.110223024625157 10−15, 315, 0]

Please see the Maxima manual for details.

87

7 Calculus

— Exercises

38. Compute the following limits:

limn→∞

n+1n−1 , lim

n→∞

(1 + 1

n

)n, lim

x→0

sin(x)x , lim

x→0

sin( 1x )

x , limx→0

sin(

1x

), lim

x→0x sin

(1x

),

limx→0

exp(

1x

), lim

x→0+exp

(1x

), lim

x→0−exp

(1x

).

39. Compute the first derivative of f (x) = x5 as the limit of the difference quotient.

40. Compute and simplifyn∑

k=0(n

k), where (nk) denotes the binomial coefficients.

(Use apropos to find a function that computes (nk).)

41. Computen∑

i=1(aibn−i+1 − an−i+1bi) for n = 10, 102, 103, 104, 105, 106.

42. Create a polynomial with roots 1, 2, 3, . . . , 999, 1000.(Compute the product of all linear factor.)

43. Compute all partial derivatives of first and second order of f (x, y) = xα yβ.

44. Compute the total derivative of log( f (x, y)).

45. We are given a function f (x, y, t). Assume that y = y(t) and x = x(y).

Compute ddt f (x, y, t) and d2

dt2 f (x, y, t). Display the derivatives both in Leibniz notation and assubscripts.

46. Compute the Jacobian matrix of the function f given by( f1(x, y), f2(x, y))′ = (exp(−x2 − y2), x2 + y2)′.

47. Compute the Taylor series of√

1 + x2 about the point a = 0.

48. Compute the Taylor polynomial of order 4 of√

1 + x2 around the point a = 0.Plot both the function and the polynomial.

49. Compute the local extrema of f (x) = x2+1x . Plot the graph of the function.

50. Compute the local extrema of f (x, y) = −x2 + xy + y2.Plot the graph of the function near its extrema.

51. Compute the local extrema of g(x, y) = 2(y− x2)2 + (1− x)2.Plot the graph of the function near its extrema.

52. Compute the local extrema of h(x, y) = 1x ln(x)− y2 + 1.

Plot the graph of the function near its extrema.

53. Compute∫

x3 sin(−x)dx.

54. Compute∫ π

0x3 sin(−x)dx.

55. Compute∫ 2

1tan(ln(x))dx numerically. Store the numerical result in variable a.

88

8 Ordinary Differential Equations (ODE)

8.1 Analytic Solutions of Ordinary Differential Equations

Command Description

ode2(eqn,dvar,ivar) Solves ODE of first and second orderic1(sol,xval,yval) Solves initial value problems of first orderic2(sol,xval,yval,dval) Solves initial value problems of second order


%c integration constant for first order ODEs%k1,%k2 integration constants for second order ODEs

System Variable Default Description

method Shows successful solution method

An ordinary differential equation is an equation where the unknown is not a number but a func-tion. First-order differential equations contain first derivatives and can be written as

y′ = f (x, y) .

Second-order differential equations contain first and second derivatives:

y′′ = f (x, y, y′) .

An example for a first-order derivative is

x2 dydx

+ 3yx =sin(x)

x

There are two ways to represent ordinary differential equations in Maxima. The simplest way is torepresent the derivatives by ’diff(y,x) and ’diff(y,x,2), respectively. The above ordinarydifferential equation would then be entered as

(%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;

(%o1) x2(

ddx

y)+ 3yx =

sin(x)x

Note that the derivative ’diff(y,x) is quoted, to prevent it from being evaluated (to 0). Thesecond way is to write y(x) explicitly as a function of x. The above equation would then be enteredas

(%i2) x^2*diff(y(x),x) + 3*y(x)*x = sin(x)/x;

(%o2) x2(

ddx

y(x))+ 3y(x)x =

sin(x)x

89


When we solve such an equation we have to find an appropriate function y(x). Routine ode2tries to solve first and second-order ODEs. It takes three arguments: an ODE given by eqn, thedependent variable dvar, and the independent variable ivar. The function tries various integra-tion methods to find an analytic solution. When successful, it returns either an explicit or implicitsolution for the dependent variable. %c is used to represent the integration constant in the caseof first-order equations. If routine ode2 cannot obtain a solution for whatever reason, it returnsfalse.

(%i3) ode2(x^2*'diff(y,x) + 3*y*x = sin(x)/x, y, x);

(%o3) y =%c− cos(x)

x3

Routine ode2 stores information about the method used to solve the differential equations in vari-able method. The above differential equation has been recognized as a linear ODE.

(%i4) method;

(%o4) linear

In the case where y(x) is given, then the independent variable must be given as function y(x).B(%i5) ode2(x^2*diff(y(x),x) + 3*y(x)*x = sin(x)/x, y(x), x);

(%o5) y(x) =%c− cos(x)

x3

Initial value problems of first order can be solved by routine ic1. It takes a general solution to theequation as found by ode2 as its first argument. The second argument xval gives an initial valuefor the independent in the form x=x0 and the third argument yval gives the initial value for thedependent variable in the form y=y0.

Thus we get the solution of the initial value problem

x2 dydx

+ 3yx =sin(x)

x, y(π) = 0

in the following way:

(%i6) sol1: ode2(x^2*'diff(y,x) + 3*y*x = sin(x)/x, y, x);

(%o6) y =%c− cos(x)

x3

(%i7) ic1(sol1, x=%pi, y=0);

(%o7) y = − cos(x) + 1x3

Similarly we can use ode2 to solve second-order differential equations. In this case the two inte-gration constants are represented by %k1 and %k2.

(%i8) sol2: ode2('diff(y,x,2) + y*'diff(y,x)^3 = 0, y, x);

(%o8)y3 + 6 ∗%k1 ∗ y

6= x + %k2

The corresponding initial value problem can be solved by means of ic2. Compared to ic1 is hasan additional forth argument dval that gives the initial value of the first derivative of the depen-dent variable with respect to the independent variable in the form diff(y,x)=dy0.

(%i9) ic2(sol2, x=0, y=0, 'diff(y,x)=2);

(%o9)y3 + 3 ∗ y

6= x

90

8.2 Linear Systems

8.2 Linear Systems

Command Description

desolve(eqn, y) Solves linear ODEdesolve([eqn_1,...,eqn_n],[y_1,..,y_n]) Solves system of linear ODEsatvalue(expr,[x_1=a_1,...],c) Assigns value c to expr at point x=a

(Systems) of linear equations can be found by routine desolve. It is important that the func-tions yi(x) must be given in functional form. The solution of the linear second-order differential Bequation

y′′(x) + y(x) = 2x

can be found as follows:

(%i1) desolve(diff(y(x),x,2)+y(x)=2*x, y(x));

(%o1) y(x) = sin(x)(

ddx

y(x)∣∣∣x=0− 2)+ y(0) cos(x) + 2 x

If desolve cannot find a solution to the given ODE, then false is returned.

If initial conditions at x = 0 are known, they can be supplied by using atvalue. Notice that theconditions can only be given at 0, and must be given before the equations are solved. It is recom- Bmended to remove your (global) assignment immediately after solving the initial value problemas otherwise it may cause obscure error messages or (even worse) wrong results.

(%i2) atvalue(y(x), x=0, 1)$

(%i3) atvalue(diff(y(x),x), x=0, 2)$

(%i4) desolve(diff(y(x),x,2)+y(x)=2*x, y(x));

(%o4) y(x) = cos(x) + 2x

(%i5) kill(y)$

Similarly systems of linear ODEs can be solved.

x′(t) = x(t)− 2y(t)y′(t) = −x(t) + y(t) (∗)

(%i6) eqn1: diff(x(t),t)=x(t)-2*y(t);

(%o6)ddt

x(t) = x(t)− 2y(t)

(%i7) eqn2: diff(y(t),t)=-x(t)+y(t);

(%o7)ddt

y(t) = y(t)− x(t)

(%i8) desolve([eqn1,eqn2],[x(t),y(t)]);

(%o8) [x(t) = %et

((2(−2y(0)− x(0)) + 2x(0))sinh(

√2t)

23/2 + x(0)cosh(√

2t)

),

y(t) = %et

((2y(0) + 2(−y(0)− x(0)))sinh(

√2t)

23/2 + y(0)cosh(√

2t)

)]

91


The initial value problem with x(0) = 1 and y(0) = 1 is then solved by

(%i9) atvalue(x(t), t=0, 1)$

(%i10) atvalue(y(t), t=0, 1)$

(%i11) desolve([eqn1,eqn2],[x(t),y(t)]);

(%o11) [x(t) = %et(cosh(√

2t)−√

2sinh(√

2t)),

y(t) = %et(cosh(√

2t)− sinh(√

2t)√2

)]

Routine desolve used the so called Laplace transform to solve the given ODE. Thus the returnedsolution may look weird and is hardly useful. This is in particular the case when the coefficients ofthe linear differential equations are not constants.

8.3 Direction Fields

Command Description

plotdf(dxdy,[x,y],opts) Plot direction field of a single ODEplotdf([dxdt,dydt],[x,y],opts) Plot direction field of a set of two autonomous ODEs


[trajectory_at,x,y] empty Starting point of an integral curve[direction,dir] both direction of the independent variable to compute an

integral curve (forward, backward, both)[versus_t,num] 0 If 1, curves are also plot as function of t[x,x_min,x_max] automatic Range for x-axis[y,y_min,y_max] automatic Range for y-axis

plotdf creates a plot of a vector field (direction field) of a first-order ODE or a system of twofirst-order ODEs. The directions field is a graphical representation of the solutions of an ODE.

8.3.1 Direction Fields of a Single First-Order ODE

To plot the direction field of a single ODE, the ODE must be written in the form

dydx

= F(x, y)

and the function F should be given as the first argument forplotdf. The name of the independentand the dependent variable are given as a list as second argument. If these are called x and y,respectively, it may be omitted.

When called plotdf opens a new window that contains a the direction plot. The length of thedraw vectors are proportional to the absolute value of the first derivative.

A direction field for the differential equation

y′ = exp(−x) + y

is created by the following Maxima code. Notice that it might be necessary to load packageplotdffirst.

(%i1) load(plotdf)$

(%i2) plotdf(exp(-x)+y)$

92


The resulting plot is shown in Fig. 8.1, l.h.s. The menu in the plot window has several options (notshown in the figure as its appearance heavily depends on computing environment) that allows tomanipulate the plot. Clicking into the direction field on particular point adds a trajectory startingat that point, see Fig. 8.1, r.h.s. It is possible to draw two or more trajectories with different startingpoints. One can also enter staring points by means of option trajectory_at.

(%i3) plotdf(exp(-x)+y, [trajectory_at,2,-0.1])$

Option direction controls the direction of the independent variable that will be followed tocompute an integral curve. Possible values are forward, to make the independent variable in-crease, backward, to make the independent variable decrease, or both that will lead to an integralcurve that extends in both directions.

It is also possible to plot the integral curve as a function of the independent variable t in a secondplot window. This can be switched on using the appropriate option in the menu of the plot windowor by means of command line option versus_t.

For further options we refer to the corresponding man page.

Figure 8.2 shows the result of the next Maxima command.

(%i4) plotdf(exp(-x)+y, [trajectory_at,2,-0.1], [direction,forward],[versus_t,1])$

-10 -5 0 5 10

-10

-5

0

5

10

y

x

-10 -5 0 5 10

-10

-5

0

5

10

y

x

Figure 8.1: Direction field without (l.h.s.) and with trajectory (r.h.s.).

8.3.2 Direction Fields of Two Autonomous First-Order ODEs

To plot the direction field of a set of two autonomous ODEs, they must be written in the form

dxdt

= F(x, y),dydt

= G(x, y)

and the the first argument for plotdf should be a list with the two functions F and G, in thatorder; namely, the first expression in the list will be taken to be the time derivative of the variablerepresented on the horizontal axis, and the second expression will be the time derivative of thevariable represented on the vertical axis. Those two variables do not have to be x and y, but if they

93


-10 -5 0 5 10

-10

-5

0

5

10

y

x

x

y

0 1 2 3 4 5 6

-10

-5

0

5

10

t

Figure 8.2: Direction field with trajectory starting at x = 2 and y = −0.1 (l.h.s.) and the plot againstindependent variable t (r.h.s.).

are not, then the second argument given to plotdfmust be another list naming the two variables,first the one on the horizontal axis and then the one on the vertical axis.

Here is the code plotting the directory field for the system (∗) in Sect. 8.2 on p. 91 above; see Fig. 8.3for the result.

(%i5) plotdf([x-2*y, -x+y], [trajectory_at,1,1], [versus_t,1])$

-10 -5 0 5 10

-10

-5

0

5

10

y

x

x

y

0 1 2 3 4 5 6

-10

-5

0

5

10

t

Figure 8.3: Direction field of two autonomous ODEs with trajectory starting at x = 1 and y = 1(l.h.s.) and the plot against independent variable t (r.h.s.).

94


— Exercises

56. Compute the general solutions of the following ordinary differential equations by means ofroutine ode2 as well as the corresponding initial value problems with initial values y(1) = 1.Determine the methods that are used to solve these ODEs.

(a) y′ − k yx = 0

(b) x y′ − (1 + y) = 0

(c) y′ = x y

(d) y′ + ey = 0

(e) y′ = y2

(f) y′ =√

x3 y

57. Compute the general solutions of the following ordinary differential equations by means ofroutine ode2 as well as the corresponding initial value problems with the given initial values.Determine the methods that are used to solve these ODEs.

(a) y′′ + y′ − 2y = 3 with y(0) = y′(0) = 1.

(b) y′′ − 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 0.

(c) y′′ + 2y′ + 17y = 0 with y(0) = 0 and y′(1) = 1.

58. Compute the general solutions of the following ordinary differential equations by means ofroutine desolve as well as the corresponding initial value problems with the given initialvalues.

(a) x y′ − (1 + y) = 0 with y(0) = 1.

(b) y′′ + y′ − 2y = 3 with y(0) = y′(0) = 1.

(c) y′′ − 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 0.

(d) y′′ + 2y′ + 17y = 0 with y(0) = 0 and y′(1) = 1.

59. Plot the direction fields and some trajectories for the following ordinary differential equa-tions.

(a) y′ − yx = 0

(b) x y′ − (1 + y) = 0

(c) y′ = x y

(d) y′ + ey = 0

(e) y′ = y2

(f) y′ =√

x3 y (Hint: use options x and y.)

60. Plot the direction fields and some trajectories for the following system of ordinary differentialequations. Use plotting range [0, 10]× [0, 10].

x′(t) = x(t)(1− y(t))y′(t) = −y(t)(1− x(t))

95

9 Economic Examples

9.1 Monopoly

Assume that a local provider has a monopoly over the provision of rock concerts. The firm hasestimated that its market demand curve can be drawn from the following equation

p = 124− 6q .

The marginal revenue (MR) in this case would be MR = 124− 12q.

(%i1) p(q):= 124-6*q;

(%o1) p(q) := 124− 6 q

(%i2) MR(q):= ''(diff(q*p(q),q));

(%o2) MR(q) := 124− 12 q

The firm’s average and marginal cost are constant, in our case let MC = AC = 40. The solution inthe single price case is found by setting MR = MC.

(%i3) MC(q):= 40;

(%o3) MC(q) := 40

(%i4) eq1: MR(q)=MC(q);

(%o4) 124− 12 q = 40

(%i5) q0: ev(q, solve(eq1,q));

(%o5) 7

So we find q̂ = 7. The computation of the optimal price p̂ and the optimal single price monopolyis then straightforward.

(%i6) p0: p(q0);

(%o6) 82

(%i7) pi(p,q):= p*q-40*q;

(%o7) π (p, q) := p q− 40 q

(%i8) pi(p0,q0);

(%o8) 294

97

9 Economic Examples

(%i9) plot2d([MR(q),p(q),MC(q)],[q,0,11],[color,green,blue,red], [legend, "MR(q)","p(q)","MC(q)"]);

(%o9)

-20

0

20

40

60

80

100

120

140

0 2 4 6 8 10

q

MR(q)P(q)

MC(q)

9.2 Profit Maximization

The notion of optimization of a function is central to economic analysis of rational behaviour.For example, firms are usually assumed to want to choose an output level that maximises profitand minimizes the cost. We solve like above an another problem of a monopolist who wants tomaximize his output.

Assume a monopoly faces a demand function given by

p(q) = 50− 12

q

and a total cost function ofTC(q) = 2 + 60 q− 8 q2 + q3 .

In order to find the maximum profit output we first need to derive the total revenue function TRand the total profit function TP in the usual way.

(%i1) p(q):= 50-1/2*q;

(%o1) p(q) := 50− 12

q

(%i2) TC(q):= 2+60*q-8*q^2+q^3;

(%o2) TC(q) := 2 + 60 q + (−8) q2 + q3

(%i3) TR(q):= ''(expand(p(q)*q));

(%o3) TR(q) := 50 q− q2

2(%i4) TP(q):=''(expand(TR(q)-TC(q)));

(%o4) TP(q) := −q3 +15 q2

2− 10 q− 2

Since this function is not concave we apply the following method: Compute the candidate pointsfor the (global) maximum and compare their total profits: the critical points of TP and the bound-ary points q = 0 (notice that q must satisfy the nonnegativity condition) and q = ∞ (that is,limq→∞ TP(q)). Notice that all critically points are feasible.

98

9.3 Price Discrimination

(%i5) cp: solve(diff(TP(q),q)=0,q);

(%o5) [q = −√

105− 156

, q =

√105 + 15

6]

(%i6) float(cp);

(%o6) [q = 0.792174872340067, q = 4.207825127659932]

(%i7) [TP(0),TP(rhs(cp[1])),TP(rhs(cp[2])),limit(TP(q),q,inf)], numer;

(%o7) [−2,−5.712313244682943, 14.21231324468293,−in f ]

Hence we have maximum profit q∗ = 4.2078. By substitution we get maximum profit and optimalprice.

(%i8) q0: rhs(cp[2]), numer;

(%o8) 4.207825127659932

(%i9) TP0: TP(q0);

(%o9) 14.21231324468293

(%i10) p0: p(q0);

(%o10) 47.89608743617003

At last we plot total profit and marginal profit as well as total revenue and total price (the later twoare rescaled for the picture).

(%i11) plot2d([TP(q),diff(TP(q),q),TC(q)/8,TR(q)/8], [q,0,6],[legend, "TP(q)", "MP(q)", "TC(q)", "TR(q)"]);

(%o11)

-30

-20

-10

0

10

20

30

40

0 1 2 3 4 5 6

q

TP(q)MP(q)TC(q)TR(q)

9.3 Price Discrimination

Assume that a provider sales a good at two different prices

p1(q1) = 20− 2 q1

p2(q2) = 10− q2

and has cost functionC(q1, q2) = (q1 + q2)

2 + 5

99

9 Economic Examples

The provider wants to maximize its profit. One approach is to define two functions p1 and p2.However, here we want to demonstrate the possibility of vector valued functions in Maxima. Thuswe need a vector q = (q1, q2) and a function p(q) =

(p1(q1, q2), p2(q1, q2)

).

(%i1) q: [q[1], q[2]];

(%o1) [q1, q2]

(%i2) p(q):= [20-2*q[1], 10-q[2]];

(%o2) p(q) := [20− 2 q1, 10− q2]

(%i3) C(q):=(q[1]+q[2])^2+5;

(%o3) C(q) := (q1 + q2)2 + 5

(%i4) TP(q):=p(q).q-C(q);

(%o4) TP(q) := p(q) · q− C(q)

In order to get the maximal profit we have to compute the stationary points of TP.

(%i5) MP(q):= ''([diff(TP(q),q[1]), diff(TP(q),q[2])]);

(%o5) MP(q) := [−2 (q2 + q1)− 4 q1 + 20,−2 (q2 + q1)− 2 q2 + 10]

(%i6) cq: solve(MP(q), q);

(%o6) [[q2 = 1, q1 = 3]]

May also may check whether the point indeed is a global maximum. Thus we show that TP isconcave by computing all eigenvalues of its Hessian.

(%i7) hessian(TP(q), q);

(%o7)

[−6 −2−2 −4

](%i8) float(eigenvalues(%));

(%o8) [[−7.23606797749979,−2.76393202250021], [1.0, 1.0]]

Hence we have optimal profit for q∗ = (3, 1).

(%i9) q0: ev(q, cq);

(%o9) [3, 1]

We also can easily verify that marginal revenues and marginal costs coincide.

(%i10) MR(q):= ''([diff((p(q)*q)[1],q[1]), diff((p(q)*q)[2],q[2])]);

(%o10) MR(q) := [20− 4 q1, 10− 2 q2]

(%i11) MC(q):= ''([diff(C(q),q[1]), diff(C(q),q[2])]);

(%o11) MC(q) := [2 (q2 + q1), 2 (q2 + q1)]

(%i12) MR(q0); MC(q0);

(%o12) [8, 8]

(%o13) [8, 8]

100

9.4 Cournot Duopoly

9.4 Cournot Duopoly

In the Cournot duopoly model we have a market with demand function

p(q1, q2) = a− b (q1, q2)

and two providers with respective cost functions

C1(q1) = d1 + c1 q1

C2(q2) = d2 + c2 q2

and profit functionsπi(q1, q2) = p(q1, q2) qi − Ci(qi) , i = 1, 2

If we use vector valued function this can be coded as follows.

(%i1) q: [q[1],q[2]]$

(%i2) p(q):= a-b*(q[1]+q[2]);

(%o2) p(q) := a− b (q1 + q2)

(%i3) c:[c[1],c[2]]$ d:[d[1],d[2]]$

(%i4) C(q):= d+c*q;

(%o4) C(q) := d + c q

(%i5) pi(q):= p(q)*q - C(q);

(%o5) π(q) := p(q) q− C(q)

Each company tries to maxize its profit given the supply of its competitor. Thus we get the socalled reaction function as

(%i6) R: [solve(diff(pi(q)[1],q[1]),q[1])[1],solve(diff(pi(q)[2],q[2]),q[2])[1]];

(%o6) [q1 = − q2 b− a + c1

2 b, q2 = − q1 b− a + c2

2 b]

Thus if each company will react on any change in supply of its opponent we only have equlibriumif both equalities q1 = − q2 b−a+c1

2 b and q2 = − q1 b−a+c22 b hold simultaniously. We then have reached

a so called Nash equlibrium because none of the two companies can change his supply withoutdecreasing its profit.

(%i7) solve(R,q);

(%o7) [[q1 =a + c2 − 2 c1

3 b, q2 =

a− 2 c2 + c1

3 b]]

9.5 Cobb-Douglas Production Function

The Cobb Douglas function is widely used to represent the relationship of an output to inputs.The function was proposed by K. Wicksell and tested against statistical evidence by C. Cobb and P.Douglas. In general the function has the following form

u = c xαyβ

whereby u describes the total production, x ≥ 0 the labor input, y ≥ 0 the capital input and c thetotal factor productivity. The exponents α > 0 and β > 0 are the output elasticities of labor andcapital.

101

9 Economic Examples

Here is example of an Cobb-Douglas function plot with α = 12 , β = 1

3 and constant c = 3.

(%i1) plot3d(3*x^(1/2)*y^(1/3), [x,0,5], [y,0,5]);

(%o1)

0 1

2 3

4 5 0

1 2

3 4

5

0

2

4

6

8

10

12

z

3*sqrt(x)*y^(1/3)

x

y

z

We can easily check that the Cobb-Douglas function is strictly concave whenever α + β < 1.

(%i2) u(x,y):= c*x^alpha*y^beta;

(%o2) u(x, y) := c xα y beta

(%i3) hess: hessian(u(x,y), [x,y]);

(%o3)

[(α− 1) α c xα−2 yβ α β c xα−1 yβ−1

α β c xα−1 yβ−1 (β− 1) β c xα yβ−2

](%i4) factor(determinant(hess));

(%o4) −α β (β + α− 1) c2 x2 α−2 y2 β−2

Since 0yα, β < 1 and 0 < α + β < 1, we find for x, y > 0 for the leading principal minors:

(α− 1) α c xα−2 yβ < 0−α β (β + α− 1) c2 x2 α−2 y2 β−2 > 0

and thus u(x, y) is strictly concave.

9.6 Stationary Points and Cobb-Douglas Function

Suppose we need the maximum of the function

f (x, y) = 6 x12 y

13 − 3 x− 2 y .

We already have seen in Sect. 9.5 that f is strictly concave, since α + β = 56 < 1. Thus it remains to

find stationary points.

102

9.6 Stationary Points and Cobb-Douglas Function

(%i1) f(x,y):= 6*x^(1/2)*y^(1/3)-3*x-2*y;

(%o1) f (x, y) := 6 x1/2 y1/3 − 3 x + (−2) y

(%i2) plot3d(f(x,y), [x,0,3], [y,0,3], [z,0,1]);

(%o2)

0 0.5 1 1.5 2 2.5 3 0 0.5

1 1.5

2 2.5

3

0

0.2

0.4

0.6

0.8

1

z

-2*y+6*sqrt(x)*y^(1/3)-3*x

x

y

z

(%i3) fx: diff(f(x,y), x);

(%o3)3 y1/3√

x− 3(%i4) fy: diff(f(x,y), y);

(%o4)2√

xy2/3 − 2

(%i5) solve([fx=0,fy=0],[x,y]);

(%o5) [ ]

Unfortunately, solve does not find a solution. However, the following trick does the job. We tellMaxima that both varaibles are positive and solve one equation and substitute into the other one.

(%i6) assume(x>0,y>0);

(%o6) [x > 0, y > 0]

(%i7) solve(fx=0,x);

(%o7) [x = y2/3]

(%i8) yo: solve(ev(fy,%)=0,y);

(%o8) [y = 1]

(%i9) xo: solve(ev(fx,yo)=0,x);

(%o9) [x = 1]

Thus the point (1, 1) maximizes f .

Now let us look at the general function

f (x, y) = c xα yβ − r x− w y

where α + β < 1. Thus f is strictly concave and we only have to find the stationary points of f .However, if we want to solve both equalities simulatiously, we only get an error message.

103

9 Economic Examples

(%i10) kill(all)$

(%i11) f(x,y):= c*x^a*y^b-r*x-w*y;

(%o11) f (x, y) := c xa yb − r x + (−w) y

(%i12) fx: diff(f(x,y), x);

(%o12) a c xa−1 yb − r

(%i13) fy: diff(f(x,y), y);

(%o13) b c xa yb−1 − w

(%i14) assume(x>0,y>0)$

(%i15) solve([fx=0,fy=0],[x,y]);algsys: tried and failed to reduce system to a polynomial in onevariable; give up.-- an error. To debug this try: debugmode(true);

Thus let us try the same trick as above. However, we have to answer some more questions.

(%i16) solve(fx=0,x);Is a an integer? no;Is a c r positive, negative, or zero? pos;

(%o16) [x =

( ra c) 1

a−1

yb

a−1]

(%i17) yo: solve(ev(fy,%)=0,y);Is (b+a-1)/(a-1) an integer? no;Is a b c^2 r w positive, negative, or zero? pos;

(%o17) [y =

w

b c(

ra c )

1a−1

)a

1

b+a−1−a

b+a−1

]

(%i18) xo: solve(ev(fx,yo)=0,x);Is a^2 b c^3 r^2 w positive, negative, or zero? pos;

(%o18) [x =

r

w

b c(

ra c )

1a−1

)a

1b+a−1−

ab+a−1

b

a c

1a−1

]

It works although the result is not very useful.

9.7 Constraint Optimization

Suppose we have to maximize a Cobb-Douglas production function

Y(K, L) = L2K

under constraintg(K, L) = K + L = 3.

104

9.7 Constraint Optimization

Such problems can be solved by means of Lagrange multipliers. Thus we have to find the station-ary points of the Lagrange function

L(K, L, λ) = L2K + λ(3− L− K)

Although we can symbol L for both the Lagrange function L and the variable L (as Maxima distin-guishes between functions and variables) it is recommended to use different symbols.

(%i1) Y(K,L):= K^2*L;

(%o1) K2 L

(%i2) g(K,l):= K+L;

(%o2) K + L

(%i3) F(K,L,mu):= ''(Y(K,L) + mu*(3-g(K,L)));

(%o3) F(L, K, µ) := K2L + µ(3− K− L)

(%i4) gradF: jacobian([F(K,L,mu)],[K,L,mu])[1];

(%o4) [2 K L− µ, K2 − µ, −L− K + 3]

(%i5) cp: solve(gradF);

(%o5) [[L = 1, K = 2, µ = 4], [L = 3, K = 0, µ = 0]]

Thus we have found two stationary points. For the next step we have to compute the borderedHessian of Y(K, L) in order to distinguishe between local maxima and local minima. Unfortu-nately, yet there is no function that computes the border Hessian and we have to do it ourselves.Recall that (in our case)

H̄ =

0 gK gLgK LKK LKLgl LLK LLL

(%i6) H: matrix([0, diff(g(K,L),K), diff(g(K,L),L)],

[diff(g(K,L),K), diff(F(K,L,mu),K,2), diff(F(K,L,mu),K,1,L,1)],[diff(g(K,L),L), diff(F(K,L,mu),L,1,K,1), diff(F(K,L,mu),L,2)]);

(%o6)

0 1 11 2 L 2 K1 2 K 0

(%i7) ev(determinant(H), cp[1]);

(%o7) 6

(%i8) ev(determinant(H), cp[2]);

(%o8) −6

(%i9) cp[1];

(%o9) [L = 1, K = 2, µ = 4]

Hence we have only one local maximum given by L = 1, K = 2 and µ = 4.

ImportantRecall that the Maxima code for second order mixed partial derivatives is Bdiff(F(K,L), L,1, K,1).

105

Answers to the Exercises

ImportantThis chapter just lists Maxima code that helps to solve the given problems. By no means these are Bcomplete answers to all of the exercises. In particular in many cases the Maxima output needsfurther interpretation.

Also notice that there is usually more than one approach to solve a problem with Maxima. Theremay be much more ways to get a sensible result to the exercises than those listed in this chapter.

1. gr: (1+sqrt(5))/2;

float(gr);

2. gr: (1+sqrt(5))/2; ev(x^2-x-1,x=gr); %, numer;

3. solve(x^2-x-1=0,x);

4. (-1)^(1/3);

solve(x^3=-1,x);

5. ev(bfloat(%pi), fpprec:100);

6. ln: log;

7. log10(x):= float(log(x)/log(10));

log10(1000);

8. heron(a,b,c):= 0.25*sqrt((a^2+b^2+c^2)^2-2*(a^4+b^4+c^4));

9. ? solve;

10. apropos("solve");

11. reset(); kill(all);

12. expand((a+b)^10);

13. factor(a^10-b^10);

factor(a^10+b^10);

14. ratsimp((a^10-b^10)/(a^2-b^2));

15. solve(x^2+2*(sin(alpha)+cos(alpha))*x+sin(2*alpha)+1=0, x);

trigexpand(%); trigsimp(%);

unique(%);

16. sgn(x):= if (x<0) then -1 else if (x=0) then 0 else 1;

sgn(-1); sgn(0); sgn(%pi);

17. assume(alpha>0,beta>0,alpha<1,beta<1,alpha+beta<1);

is(alpha>-1/2);

is(alpha> 1/2);

is(alpha> 1);

forget(alpha>0,beta>0,alpha<1,beta<1,alpha+beta<1);

107


18. numeric(x,prec):= ev(bfloat(x), fpprec:prec);

numeric(1/%e,30);

19. plot2d(1/sqrt(2*%pi)*exp(-x^2/2),[x,-3,3],[ylabel,"density"]);

20. plot2d([1/sqrt(2*%pi)*exp(-x^2/2), 1/(%pi*(1+x^2))], [x,-3,3],[legend, "Gaussian distribution", "Cauchy distribution"], [style,lines, [lines, 5]], [color, black, red], [ylabel,"density"]);

21. plot2d([1/x,1/x^2,1,x,x^2],[x,0,2],[y,0,2]);

22. datalist: [[.25,-1.03], [.5,-0.63], [.75,-0.28], [1.,0.],[1.25,0.22], [1.5,0.38], [1.75,0.47]];

plot2d([log(x), [discrete, datalist]], [x,0,2], [y,-2,1], [style,lines, points], [point_type, diamond]);

23. plot2d([parametric, sin(4*%pi*t), cos(4*%pi*t)],[t,0,4],[nticks,401]);

24. /*a*/ plot3d(1/(2*%pi)*exp(-0.5*(x^2+y^2)), [x,-3,3], [y,-3,3]);

/*b*/ plot3d(1/(2*%pi)*exp(-0.5*(x^2+y^2)), [x,-3,3], [y,-3,3],[palette,false], [color, blue]);

/*c*/ plot3d(1/(2*%pi)*exp(-0.5*(x^2+y^2)), [x,-3,3], [y,-3,3],[palette,false], [color, cyan, blue]);

/*d*/ plot3d(1/(2*%pi)*exp(-0.5*(x^2+y^2)), [x,-3,3], [y,-3,3],[mesh_lines_color,false]);

25. load(draw)$

draw2d(ylabel="density", explicit(1/sqrt(2*%pi)*exp(-x^2/2),x,-3,3));

26. load(draw)$

draw2d(ylabel="density", key="Gaussian distribution", color=black,explicit(1/sqrt(2*%pi)*exp(-x^2/2), x,-3,3), key="Cauchydistribution", color=red, line_width=5, explicit(1/(%pi*(1+x^2)),x,-3,3));

27. load(draw)$

draw2d(yrange=[0,2], key="1/x", color=blue, explicit(1/x,x,0,2), key="1/x^2", color=red, explicit(1/x^2, x,0,2), key="1",color=green, explicit(1, x,0,2), key="x", color=magenta, explicit(x,x,0,2), key="x^2", color=black, explicit(x^2, x,0,2));

28. load(draw)$

datalist: [[.25,-1.03], [.5,-0.63], [.75,-0.28], [1.,0.],[1.25,0.22], [1.5,0.38], [1.75,0.47]];

draw2d(yrange=[-2,1], color=blue, explicit(log(x), x,0,2),color=red, point_type=filled_diamant, point_size=2,points(datalist));

29. load(draw)$

draw2d(nticks=29, parametric(sin(4*%pi*t),cos(4*%pi*t), t,0,4));

30. load(draw)$

draw3d(explicit(1/(2*%pi)*exp(-0.5*(x^2+y^2)), x,-3,3, y,-3,3));

31. x: [1,2,3]; y: [4,5,6];

x+y; x-y; x.y; 3*x;

108

32. A: matrix([1,2,3],[4,5,6],[7,8,9]);

B: matrix([-1,0,2],[-2,1,3],[2,4,-1]);

A+B; A-B; 3*A; A.B; A^^2; B^^(-1);

determinant(A); transpose(A);

A^^(-1);

33. A: matrix([1,2,3],[4,5,6],[7,8,9]);

B: matrix([-1,0,2],[-2,1,3],[2,4,-1]);

rank(A); rank(B);

nullity(A); nullity(B);

nullspace(A); nullspace(B);

columnspace(A); columnspace(B);

CS: columnspace(A); a1*first(CS)+a2*second(CS);

34. "(a)"

eq1: 2*x[1]+3*x[2]+4*x[3]= 2;

eq2: 4*x[1]+3*x[2]+ x[3]=10;

eq3: x[1]+2*x[2]+4*x[3]= 5;

solve([eq1,eq2,eq3], [x[1],x[2],x[3]]);

"(b)"

eq1: 2*x[1]+2*x[2]+ x[3]+3*x[4]=10;

eq2: 3*x[1]+5*x[2]+2*x[3]- x[4]=30;

eq3: x[1]+2*x[2]+ x[3]- x[4]=12;

solve([eq1,eq2,eq3], [x[1],x[2],x[3],x[4]]);

"(c)"

eq1: 2*x[1]+10*x[2]+4*x[3]+ 9*x[4]= 1;

eq2: x[1]+ 6*x[2]+5*x[3]+ 3*x[4]= 1;

eq3: 2*x[1]+16*x[2]+9*x[3]+11*x[4]=-1;

eq4: x[1]+ 5*x[2]+2*x[3]+ 5*x[4]= 2;

eq5: x[2]+3*x[3] = 4;

solve([eq1,eq2,eq3,eq4,eq5], [x[1],x[2],x[3],x[4]]);

35. A: matrix([1,2],[3,4]);

es: eigenvectors(A);

eigval: es[1][1];

v[1]: es[2][1]; v[2]: es[2][2];

36. A: matrix([1,2,1],[2,1,3],[1,3,1]);

es: eigenvectors(A);

eigvals: float(realpart(es[1][1]));

eigvects: float(realpart(es[2]));

109


37. A: diagmatrix(3,1);

esA: eigenvectors(A);

numberofeigenvaluesA: length(esA[1][1]);

algebraicmultiplicityA: esA[1][2];

geometricmultiplicityA: length(esA[2][1]);

B: matrix([1,1,1],[0,1,1],[0,0,1]);

esB: eigenvectors(B);

numberofeigenvaluesB: length(esB[1][1]);

algebraicmultiplicityB: esB[1][2];

geometricmultiplicityB: length(esB[2][1]);

38. limit((n+1)/(n-1),n,inf);

limit((1+1/n)^n,n,inf);

limit(sin(x)/x,x,0);

limit(sin(1/x)/x,x,0);

limit(sin(1/x),x,0);

limit(sin(1/x)*x,x,0);

limit(exp(1/x),x,0);

limit(exp(1/x),x,0,plus);

limit(exp(1/x),x,0,minus);

39. limit(((x+h)^5-x^5)/h,h,0);

40. sum(binomial(n,k),k,0,n), simpsum;

41. sum(a[i]*b[n-i+1]-a[n-i+1]*b[i],i,1,n), n:10;

42. product((x-y),y,1,1000);

43. diff(xâlpha*y^beta,x); diff(xâlpha*y^beta,y);

diff(xâlpha*y^beta,x,2); diff(xâlpha*y^beta,y,2);

diff(xâlpha*y^beta,x,1,y,1);

44. diff(log(f(x)));

45. depends(f,[x,y,t]); depends(x,y); depends(y,t);

diff(f,t);

diff(f,t,2);

derivabbrev: true$

diff(f,t);

diff(f,t,2);

46. jacobian([exp(-x^2-y^2), x^2+y^2],[x,y]);

47. powerseries(sqrt(1+x^2),x,0);

48. f(x):= sqrt(1+x^2);

t(x):= ''(taylor(f(x),x,0,4));

plot2d([f(x),t(x)], [x,-1.5,1.5]);

110

49. f(x):= (x^2+1)/x;

fx: diff(f(x),x);

fxx: diff(f(x),x,2);

crit: solve(fx,x);

ev(fxx, crit[1]);

ev(fxx, crit[2]);

plot2d(f(x), [x,-2,2],[y,-5,5]);

50. f(x,y):= -x^2+x*y+y^2;

fx: diff(f(x,y), x);

fy: diff(f(x,y), y);

crit: solve([fx,fy], [x,y]);

H: hessian(f(x,y), [x,y]);

eigenvalues(ev(H,crit[1]));

plot3d(f(x,y), [x,-1,1],[y,-1,1]);

51. g(x,y):= 2*(y-x^2)^2+(1-x)^2;

gx: diff(g(x,y), x);

gy: diff(g(x,y), y);

crit: solve([gx,gy], [x,y]);

H: hessian(g(x,y), [x,y]);

eigenvalues(ev(H,crit)), numer;

plot3d(g(x,y), [x,0.5,1.5],[y,0.5,1.5]);

52. h(x,y):= 1/x*log(x)-y^2+1;

hx: diff(h(x,y), x);

hy: diff(h(x,y), y);

crit: [solve(hx,x), solve(hy,y)];

H: hessian(h(x,y),[x,y]);

eigenvalues(ev(H,crit));

plot3d(h(x,y), [x,1,5],[y,-1,1]);

53. integrate(x^3*sin(-x),x);

54. integrate(x^3*sin(-x),x,0,%pi);

55. qres: quad_qags(tan(log(x)),x,1,2);

a: qres[1];

56. sola: ode2('diff(y,x)-k*y/x=0, y, x); ic1(sola, x=1, y=1); method;

solb: ode2(x*'diff(y,x)-(1+y)=0, y, x); ic1(solb, x=1, y=1); method;

solc: ode2('diff(y,x)=x*y, y, x); ic1(solc, x=1, y=1); method;

sold: ode2('diff(y,x)+%e^y=0, y, x); ic1(sold, x=1, y=1); method;

sole: ode2('diff(y,x)=y^2, y, x); ic1(sole, x=1, y=1); method;

solf: ode2('diff(y,x)=sqrt(x^3*y), y, x); ic1(solf, x=1, y=1);method;

111


57. sola: ode2('diff(y,x,2)+'diff(y,x)-2*y=3, y, x); method;

ic2(sola, x=0,y=1,'diff(y,x)=1);

solb: ode2('diff(y,x,2)-6*'diff(y,x)+9*y=0, y, x); method;

ic2(solb, x=0,y=2,'diff(y,x)=0);

solc: ode2('diff(y,x,2)+2*'diff(y,x)+17*y=0, y, x); method;

ic2(solc, x=0,y=0,'diff(y,x)=1);

58. sola: desolve(diff(y(x),x)-(1+y(x))=0, y(x));

atvalue(y(x), x=0, 1); desolve(diff(y(x),x)-(1+y(x))=0, y(x));

kill(y)$

solb: desolve(diff(y(x),x,2)+diff(y(x),x)-2*y(x)=3, y(x));

atvalue(y(x), x=0, 1); atvalue(diff(y(x), x), x=0, 1);

desolve(diff(y(x),x,2)+diff(y(x),x)-2*y(x)=3, y(x));

kill(y)$

solc: desolve(diff(y(x),x,2)-6*diff(y(x),x)+9*y(x)=3, y(x));

atvalue(y(x), x=0, 2); atvalue(diff(y(x),x), x=0, 0);

desolve(diff(y(x),x,2)-6*diff(y(x),x)+9*y(x)=3, y(x));

kill(y)$

sold: desolve(diff(y(x),x,2)+2*diff(y(x),x)+17*y(x)=3, y(x));

atvalue(y(x), x=0, 0); atvalue(diff(y(x),x), x=0, 1);

desolve(diff(y(x),x,2)+2*diff(y(x),x)+17*y(x)=3, y(x));

kill(y)$

59. /*a*/ plotdf(y/x);

/*b*/ plotdf((1+y)/x);

/*c*/ plotdf(y*x);

/*d*/ plotdf(-%e^y);

/*e*/ plotdf(y^2);

/*f*/ plotdf(sqrt(x^3*y),[x,0.001,10],[y,0.001,10]);

60. plotdf([x*(1-y),-y*(1-x)],[x,0,3],[y,0,3]);

112

Index

* (operator), 5, 65, 66+ (operator), 5, 65, 66- (operator), 5, 65, 66. (operator), 65, 66/ (operator), 5/*...*/ (other), 18: (operator), 7:= (operator), 7; (operator), 3< (operator), 26<= (operator), 26= (operator), 15, 26> (operator), 26>= (operator), 26? (operator), 9?? (operator), 9[...] (operator), 14# (operator), 26$ (operator), 3% (operator), 9%c (constant), 89%e (constant), 5%i (constant), 5%i1 (operator), 3%k1,%k2 (constant), 89%o1 (operator), 3, 9%pi (constant), 5^ (operator), 5^^ (operator), 66"..." (constant), 15' (operator), 24'' (operator), 24

abs (function), 6allroots (function), 15and (operator), 26append (function), 14apropos (function), 9assume (function), 27atvalue (function), 91

bfloat (function), 12

charpoly (function), 71color (option), 40, 46, 60colorbox (option), 52columnspace (function), 68

contour (option), 56contour_levels (option), 56contour_plot (function), 38cos (function), 6

declare (function), 27del (function), 77delete (function), 14denom (function), 22dependencies (variable), 78depends (function), 78derivabbrev (variable), 78desolve (function), 91determinant (function), 67diagmatrix (function), 67diff (function), 77direction (option), 92draw (function), 45draw2d (function), 45draw3d (function), 45

eigenvalues (function), 71eigenvectors (function), 71else (operator), 27enhanced3d (option), 52, 60equal (function), 26ev (function), 10, 29exp (function), 6expand (function), 22explicit (function), 46, 52

factor (function), 22facts (function), 27false (constant), 26file_name (option), 60first (function), 68float (function), 12forget (function), 27fpprec (variable), 10fpprintprec (variable), 10fullratsimp (function), 22functions (variable), 7

gr2d (function), 45gr3d (function), 45grid (option), 39, 59

hessian (function), 83

113

Index

ic1 (function), 89ic2 (function), 89ident (function), 67if (operator), 27imagpart (function), 17implicit (function), 48, 54implicit_plot (function), 36ind (constant), 75inf (constant), 5infinity (constant), 75integrate (function), 86invert (function), 67ip_grid (option), 59ip_grid_in (option), 59is (function), 26

jacobian (function), 80

key (option), 51, 58kill (function), 7

label (function), 58last (function), 68legend (option), 38length (function), 14, 68lhs (function), 15limit (function), 75line_type (option), 60line_width (option), 60load (function), 18log (function), 6

matrix (function), 66mesh_lines_color (option), 40method (variable), 89minf (constant), 5

niceindices (function), 81not (operator), 26nticks (option), 34, 39, 48, 59nullity (function), 68nullspace (function), 68num (function), 22numer (variable), 13

ode2 (function), 89or (operator), 26

palette (option), 40, 60parametric (function), 48, 54parametric_surface (function), 54plod2d (function), 32, 33, 34, 35plod3d (function), 36, 37plotdf (function), 92point_size (option), 49, 60point_type (option), 40, 49, 60

points (function), 49, 57points_joined (option), 49polarform (function), 17powerseries (function), 81

quad_qags (function), 86

radcan (function), 22rank (function), 68ratexpand (function), 22ratsimp (function), 22realpart (function), 17rectform (function), 17reset (function), 10rest (function), 68rhs (function), 15

second (function), 68set_display (function), 12set_draw_defaults (function), 58simpproduct (variable), 76simpsum (variable), 76sin (function), 6solve (function), 15sqrt (function), 6style (option), 33, 39sum (function), 76surface_hide (option), 52, 60

t (option), 34tan (function), 6taylor (function), 81terminal (option), 60then (operator), 27title (option), 58trajectory_at (option), 92transpose (function), 67trigreduce (function), 22trigsimp (function), 17, 22true (constant), 26

und (constant), 75unknown (constant), 26

values (variable), 7versus_t (option), 92

x (option), 32, 39, 59, 92xaxis (option), 59xlabel (option), 38, 58xrange (option), 46xtics (option), 58xu_grid (option), 59

y (option), 32, 39, 59, 92yaxis (option), 59

114

Index

ylabel (option), 38, 58yrange (option), 46ytics (option), 58yv_grid (option), 59

z (option), 39, 59zaxis (option), 59zlabel (option), 38, 58ztics (option), 58

115

Introduction to Maxima for Economicsleydold/maxima/MaximaSkript.pdf · 2011. 9. 26. · 2.3 Maxima...

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Transcript of Introduction to Maxima for Economicsleydold/maxima/MaximaSkript.pdf · 2011. 9. 26. · 2.3 Maxima...